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From the parametric amplification in Electric Force
Microscopy to the Scanning Gate Microscoppy of
Quantum Rings
Frederico Martins
To cite this version:
Frederico Martins. From the parametric amplification in Electric Force Microscopy to the Scanning
Gate Microscoppy of Quantum Rings. Physics [physics]. Université Joseph-Fourier - Grenoble I, 2008.
English. �tel-00260860�
HAL Id: tel-00260860
https://tel.archives-ouvertes.fr/tel-00260860
Submitted on 5 Mar 2008
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THÈSE
pour obtenir le grade de
Docteur de l’Université Joseph Fourier - Grenoble 1
Spécialité : Physique / Physique des Materiaux
par
Frederico MARTINS
présentée et soutenue publiquement le 4 Février 2008
De l’amplification
paramétrique en microscopie à
force électrique à la
microscopie à grille locale
d’anneaux quantiques
Institut Néel - CNRS / UJF
Composition du jury:
Présidente:
Rapporteurs:
Margarida Godinho
Giancarlo Faini
Thierry Mélin
Examinateur:
Kunteah Khang
Invité:
Benoit Hackens
Directeurs de thèse: Joël Chevier
Serge Huant
2
Acknowledgment
This manuscript marks the end of a period of four years of my personal
research work. It also marks the end of my stay in Grenoble. During
this time I met many people that, for a reason of space, I cannot thank
individually. Most of these people I will, most probably, never see
again. Therefore, I think it is fair to start my acknowledgments by
thanking them. They were in many ways my family and my support
away from Portugal.
I also have to pay tribute to this city and to this country. The
unique young international environment can only be possible in a city
like Grenoble and I think that the best way to thank this country is to
confess that part of me is now french.
During the period of my thesis I had the pleasure to be part of a
team and to work with several people, with whom I obviously share
the results. I strongly believe that the good outcome was a result of
the excellent relationship within the group. In this section, I will thank
these people individually.
The first responsible for this project are the leaders of the group,
and simultaneously my thesis advisors, Serge Huant and Joël Chevrier.
I wish to thank them for offering me the opportunity to work in such a
good environment and for their constant support. In particular, I wish
to thank Joël for oppening the door when I first came to Grenoble. To
Serge, I wish to thank his patience, his daily guidance and, specially,
all the advices and encouragement during the difficult periods.
I am also very grateful to all the people I worked with.
In the beginning of my thesis I worked with Martin Stark. During
this period we set the basis of the low temperature microscope. His
initial input was fundamental. I thank his friendship, his generosity
and his scientific advices.
I also want to thank Thierry Ouisse for his constant scientific support, his culture and his ‘difficult’ questions. Most of our results were
obtained while ‘fighting’ his questions. Moreover, frequently, it was
Thierry himself bringing the solution to the ‘difficult’ questions.
Many thanks to Jean-François Motte. When nothing works, or
3
when you do a big mistake, there is always Jeff to save you. His constant
good mood, his experience and his dynamism are priceless. He also
contributed a lot to the design and to the assembling of the microscope.
Most of the experiments presented here (and also the best results)
were done together with Benoit Hackens. We spent (alone) several
months working over the night and weekends. During these nights
we debated many subjects (not always scientific) and we built a solid
friendship. If you can chose one person to work with, that person
is Benoit. He is the perfect partner. I thank Benoit for all that he
contributed to this thesis.
I would also like to thank Vincent Bayot for his brainstorms and
for his very good advises, Marco Pala that never hesitated to collaborate with us and my office mate Hermman Sellier for the scientific
debates. The substrate used in our experiments were kindly offered by
the research group from the IEMN lab (Lille), Xavier Wallart, Sylvain
Bollaert and Alain Cappy.
I am also very grateful to the other students and post-docs that I
crossed paths with during my PhD: Michael Nasse, Nicolas Chevalier,
Aurellien Cuche, Paolo Actis (l’Italien irreductible), Florian Habrard,
Wilfrid Schwartz, Alexis Mosset, Alessandro Siria and Guillaume Jourdan.
Spectial thanks is due to Yannic Sonnefraud, Mario Rodrigues,
Michal Hrouzek and Miguel Silveira that are much more than colleagues. They are good friends.
At last, I would like to thank the members of the jury Prof. Margarida Godinho, Dr. Giancarlo Faini, Dr. Thierry Mélin, Prof. Kuntheak
Kheng and Benoit Hackens for having accepted the task. To Prof. Godinho I also want to thank for sending me to Grenoble in the first place.
I could not finish without thanking to my very best friend Rodrigo,
that is always there no matter what, and to my family. To my family
there is not much to say because they know me. Nothing would possible
without them. Even far away from home they always care, and I cannot
hide that I only feel home when I am with them.
I wish to dedicate this thesis to my parents, my sister, my grandmother and my beautiful nephew.
4
Résumé
La réduction de taille des dispositifs électroniques apporte de nouvelles
exigences scientifiques et techniques. De nouveaux phénomènes apparaissent aux petites échelles et, par ailleurs, l’exploration des propriétés
électroniques à l’échelle locale nécessite le développement d’instruments
adaptés. Ces deux demandes sont devenues cruciales pour le développement de la ‘nano-électronique’.
L’objectif de cette thèse est double : augmenter la sensibilité de détection de charges déposées sur des surfaces et l’imagerie dans l’espace
réel des fonctions d’onde électroniques dans des nano-dispositifs enterrés sous la surface libre. Pour atteindre ces objectifs, nous avons
concu un microscope à force atomique (AFM) idoine. Ce microscope
est décrit dans le premier chapitre de ce mémoire.
Dans un deuxième chapitre, nous décrivons une méthode d’amplification
paramétrique pour augmenter la sensibilité de détection de charges déposées sur une surface. Le mouvement du micro-levier AFM est déterminé analytiquement et est confirmé tant par une approche numérique
que par l’expérience. Nous concluons qu’avec notre méthode, la limite
de bruit thermique peut etre dépassée. Dans le meme chapitre, nous
faisons une remarque sur une variante très répandue de la microscopie
à force électrique (EFM) : la microscopie à force de Kelvin (KFM).
Nous montrons que, meme si elle n’est pas volontairement provoquée,
l’amplification paramétrique du mouvement du micro-levier est toujours présente et qu’elle peut notablement modifier la résolution du
microscope telle qu’anticipée à travers les approches usuelles.
Dans le dernier chapitre, nous nous intéressons au transport électronique dans des systémes mésoscopiques fabriqués à partir de gaz
électroniques 2D. Traditionnellement, l’étude de ce transport est appréhendée par des mesures de conductance à 4 points en fonction de la
température. Cette approche procure une information moyennée sur
toute la taille du dispositif et, donc, perd l’information locale. Ici, nous
complétons cette approche par des mesures dans lesquelles la pointe du
microscope AFM est polarisée électriquement de sorte à perturber localement le potentiel vu par les porteurs de charge. Le balayage de
5
cette pointe au dessus du dispositif permet d’en construire une image
de sa conductance. Cette technique est appelée ‘microscopie locale à
grille ajustable’ et est désignée par son acronyme anglo-saxon : SGM.
Nous avons ici étudié un système modèle, siège d’interférences quantiques de type Aharonov-Bohm : des anneaux quantiques fabriqués à
partir d’hérérostructures à base de GaInAs. Nous couplons nos expériences à des simulations en mécanique quantique et montrons comment
la microscopie SGM permet de sonder le transport cohérent et d’imager
les fonctions d’onde dans ces anneaux.
6
Abstract
The continuous size reduction of electronic devices have brought new
technical and scientific demands. Firstly, because new peculiar physical
effects appear at small scales. Secondly, probing electronic properties
at the local scale requires new adapted instrumentation. These two
issues have become crucial to the development of the so-called nanoelectronics.
The objective of this thesis is two fold: enhancing the sensitivity of
charge detection deposed over surfaces and the real-space imaging of
the wave-function inside buried open nano-devices. To achieve these
goals we have conceived a low temperature Atomic Force Microscope
(AFM) adapted to study electrical properties over surfaces. In the first
chapter of this thesis we describe the operation of the AFM and the
technical options.
In the second chapter, we describe a parametric method to increase
the sensitivity of an AFM to deposed charges over a surface. The movement of the AFM probe is described analytically which is confirmed by
numerical solutions and experiments. We conclude that with such a
method the thermal noise limit can be beaten. In the same chapter,
we make a remark concerning a widespread technique: the Kelvin Force
Microscopy (KFM). We show that, in this case, and even if it is not intentional, parametric effect is always present which might substantially
change the expected resolution calculated from classical approaches.
In the third and last chapter, we address the electronic transport
in mesoscopic systems fabricated from two-dimensional electron gases
(2DEGs). Traditionally, this kind of samples are characterized with
four-point conductance measurements at low temperature. This technique provides information which is averaged over the size of the whole
device and, as such, losses the local information. Here, we complement
this analysis using the AFM probe as a polarized moving gate that induces a local perturbation of the potential experienced by the 2DEG.
As the tip is scanned over the surface, a conductance map is built. This
technique is called Scanning Gate Microscopy (SGM). So far, only a
limited number of SGM experiments were performed. Here, we use
7
a model sample fabricated from InGaAs heterostructure: a quantum
ring (QR). By coupling experiments and quantum mechanical simulations we conclude that SGM permits probing the coherent transport
and imaging the electronic probability density inside the QR.
8
Contents
1 Introduction
11
2 The Low Temperature Atomic Force Microscope
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Principles of Atomic Force Microscopy . . . . . . . . .
2.2.1 Force Measurement . . . . . . . . . . . . . . . .
2.2.2 Approach-Retract Curves and Topography Images
2.2.3 Fundamental Limits . . . . . . . . . . . . . . .
2.3 Conception of the Low Temperature Atomic Force Microscope . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Requirements and Overview . . . . . . . . . . .
2.3.2 Microscope Head and Optical Detection . . . .
2.3.3 Cryostat and Mechanical Isolation . . . . . . . .
2.4 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Cooling of the Microscope . . . . . . . . . . . .
2.4.2 Calibration . . . . . . . . . . . . . . . . . . . .
2.5 Summary and perspectives . . . . . . . . . . . . . . . .
15
15
17
17
20
22
3 Electric Force Microscopy in Parametric Amplification
Regime
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Determination of the instability Domains of the Cantilever Oscillation . . . . . . . . . . . . . . . . . . . . .
3.2.1 Simple approach . . . . . . . . . . . . . . . . .
3.2.2 Numerical approach . . . . . . . . . . . . . . .
3.3 Cantilever Oscillation with ωel = 2 (ω0 + ∆ω) . . . . .
3.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Experiments . . . . . . . . . . . . . . . . . . . .
3.3.3 Sensitivity and Thermomechanical Noise . . . .
3.4 Parametric effects in Kelvin force microscopy . . . . . .
3.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Experiments . . . . . . . . . . . . . . . . . . . .
9
24
24
25
30
30
30
32
34
37
37
41
41
45
49
49
59
61
63
63
67
Contents
3.5
Summary and Conclusions . . . . . . . . . . . . . . . .
4 Scanning Gate Microscopy of Quantum Rings
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2 Electronic Interferences inside a Mesoscopic Structure
4.2.1 Aharonov-Bohm effect . . . . . . . . . . . . .
4.3 Scanning Gate Microscopy . . . . . . . . . . . . . . .
4.4 Experiment . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Quantum Ring . . . . . . . . . . . . . . . . .
4.4.2 Experimental Setup . . . . . . . . . . . . . . .
4.5 Experimental results . . . . . . . . . . . . . . . . . .
4.5.1 SGM image Filtering . . . . . . . . . . . . . .
4.5.2 Fringes analysis . . . . . . . . . . . . . . . . .
4.5.3 Electrical AB effect in the SGM images . . . .
4.5.4 Imaging electronic wave function . . . . . . .
5 Conclusion
10
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69
71
71
73
74
77
79
79
82
87
88
91
95
99
105
Chapter 1
Introduction
Electronic devices are everywhere in our daily life. We can find them
inside complex tools like mobile phones or laptops, but also in simple
instruments such as home digital thermometers, cloths irons or ovens.
Nowadays, industries fight hard to reduce their size.
Nevertheless, it is my experience that these technological and scientific efforts are not quite understood by most of the people. Indeed,
if we open, e.g., a laptop we realize that most of the space is taken
by the hard disc, the battery, the keyboard and the screen. A less
attentive regard would suggest that small circuits are only relevant in
applications where dimensions are critical.
This common belief is not accurate, the truth is that smaller circuits
consume, in general, less energy and are faster. These two reasons are
at least more important than the compactness of the circuits. Indeed,
at the end of the last century we became aware that traditional sources
of energy not only pollute but are also limited. Nowadays, the scientific
community is very active to find solutions. Today it is believed that
the only way to keep our lifestyle is to find new renewable clean sources
of energy and to build more efficient tools. By efficient we mean doing
more operations while consuming less energy.
Therefore, modern electronic nano-devices are dealing with an ever
reduced number of current carrying and interacting electrons. The
experimental assessment of the number of charges transiting through
structures, the localisation of their wave-functions and the manipulation and detection of single charges, all constitute important issues with
respect to the long-term development of nano-electronics. The study
of their quantum-coherent behaviour is of particular importance since
it is anticipated to provide new concepts and new designs of electron
devices.
This thesis is dedicated to provide new methods to analyze elec11
Chapter 1. Introduction
tronic nano-circuits. We have addressed two subjects: enhancing the
sensitivity of charge detection and real-space imaging of the wavefunction inside buried open nano-devices. To achieve this goal, we have
established a low temperature atomic force microscope (AFM) where
the local metallic nano-probe permits direct access to the local electronic properties. Several examples exist in literature demonstrating
the AFM performances concerning charge detection [1][2]. As explained
in Chapter 1, we have re-designed the head of a commercial low temperature AFM system [3][4] to be adapted to our experimental needs.
In this chapter, we start by explaining the general operation mode of
an AFM and the theoretical limitations. Afterwards we present our
experimental setup and the modified low temperature AFM. The options are discussed and we analyze its performance. This system is at
the heart of the research work presented afterwards.
In Chapter 2 we propose a method to beat the thermal noise limit
while detecting excess charges deposed over a surface. This method
consists in using a parametric excitation to increase the sensitivity of
fixed charges over samples. This effect is not new. It has been known
for centuries, e.g. the Botafumeiro in Santiago de Compostela (13th
century) [5], and it has been used in several scientific domains, e.g [6].
Here the originality is to make use of this effect to detect charges. We
propose an analytical model to describe the movement of the cantilever
when the tip probes electrostatic forces under the effect of parametric
excitation. These results are crosschecked by numerical solutions and
confirmed by experiments. In this chapter we discuss the value of the
parameters in which the system should be set so that one can be in the
best measurement conditions.
Later, in the same chapter, we make a remark concerning a common
method used in charge detection: the Kelvin force microscopy (KFM).
We show that in this technique the parametric effect is always present
and can even become dominant. Moreover, we found that setting it in
certain conditions one can either profit from the parametric effect or
lose sensitivity. The complete understanding is therefore important and
we conclude that, in most situations, this effect cannot be neglected.
In Chapter 3 we focus on mesoscopic samples based on two-dimensional
electron gases (2DEGs) buried under surfaces. Here, we use the AFM
probe as a flying gate to locally perturb the electronic current flowing
through the device. This recent research technique is know in literature as Scanning Gate Microscopy (SGM). For the time being, only a
few number of devices were studied and several aspects remain to be
explored. Furthermore, up to now no analysis has been developed permitting a correspondence between features observed in SGM imaging
12
and the electron wavefunctions inside a nanostructure. In this chapter,
our aim is to establish such a correspondence combining experiment,
numerical modeling and physical analysis.
To accomplish this objective we have chosen a model sample: a
quantum ring (QR), in our case patterned from a InGaAs/InAlAs heterostucture. In this family of devices several experiments have evidenced interferences formed by the electron waves propagating through
their arms [7][8][9]. In these experiments, these wave-like properties are
probed by macroscopic quantities (typically the overall current) and the
interference pattern is revealed by the subtle changes induced by, e.g.,
applying an external magnetic field or a gate voltage to modulate the
electron concentration. In general, even when they are of a spectroscopic nature, transport experiments give information on the electron
energy levels rather than direct information about the wave functions
or electron probability density in real space [10].
In the same chapter we complete this analysis using SGM. In our
experimental images we observed two different perturbation regimes.
The first regime corresponds to a region where the tip is ‘far’ from the
QR. In this case, the fine perturbing potential induces a phase shift of
the electron wavefunction. This effect is responsible for the appearance
of concentric fringes in our experimental images.
The second experimental situation is obtained when the tip is over
the QR. In these circumstances, radial fringes are observed. We show
from the comparison between experimental results and quantum mechanical simulations that these fringes reveal the electron probability
density inside the QR. This result is most relevant because it can provide new insight into the behaviour of electrons in mesoscopic devices
and it opens the perspective to design new devices.
13
Chapter 1. Introduction
14
Chapter 2
The Low Temperature Atomic
Force Microscope
Contents
2.1
Introduction
. . . . . . . . . . . . . . . . . . . . .
15
2.2
Principles of Atomic Force Microscopy . . . . .
17
2.3
2.4
2.5
2.1
2.2.1
Force Measurement . . . . . . . . . . . . . . . . . . 17
2.2.2
Approach-Retract Curves and Topography Images
2.2.3
Fundamental Limits . . . . . . . . . . . . . . . . . 22
20
Conception of the Low Temperature Atomic Force
Microscope . . . . . . . . . . . . . . . . . . . . . .
24
2.3.1
Requirements and Overview . . . . . . . . . . . . . 24
2.3.2
Microscope Head and Optical Detection . . . . . . 25
2.3.3
Cryostat and Mechanical Isolation . . . . . . . . . 30
Testing . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4.1
Cooling of the Microscope . . . . . . . . . . . . . . 30
2.4.2
Calibration . . . . . . . . . . . . . . . . . . . . . . 32
Summary and perspectives . . . . . . . . . . . . .
34
Introduction
Gaining local information has been an old ambition in physics, in particular, in solid-state physics. Indeed, a large part of the scientific community has been asking for a long time how the behaviour of ‘small’
part contributes to the general behaviour of a whole ensemble.
Many theories use bottom-up approaches, i.e. they start with the
behaviour of a single element and end up with the behaviour of a whole
15
Chapter 2. The Low Temperature Atomic Force Microscope
system. The experimental verification is always a challenge. Add to
this the fact that in the real world, impurities, defects and imperfections
can be, at the best, reduced to small quantities. But even at these
small quantities their effects can be dramatic [11] and the question
remains how is each element going to contribute and be averaged in
the behaviour of the whole device?
To address this issue an experimental field in solid-state physics
emerged: near-field microscopy. Ideally, the approach is to study directly each single element, impurity, imperfection present on a surface.
The goal is to study the morphology and the local properties on surfaces and devices on surfaces.
This family of experimental instruments is centered on a probe of
‘very small’ dimensions that is approached very close to the surface to
probe, to perturb or to manipulate the surface. As the probe comes
in situ to study some properties it should be adapted to the ‘natural’ ambient of the sample or to the ‘natural’ ambient of the physical
properties under study.
During this thesis a near-field microscope was conceived to be adapted
to study the electrical properties of buried semiconductor mesostructures. In our case we built a low-temperature atomic force microscope
(AFM) to work in the electric regime (also known in literature as electric force microscope or scanning gate microscope) [12]. This family of
instruments have been shown to be very well adapted to probe local
electronic properties thanks to the localized electric field created by the
sharp probe (e.g. [13][14][15][16][17]).
We have chosen to work at low temperatures and under magnetic
field, because these kind of structures exhibit many interesting quantum phenomena in these conditions. Working at low temperature
brings two additional advantages.
Firstly, the thermal noise is deq
creased by a factor of 300/4 ≃ 8.7, which implies a gain in force
sensitivity [18] and, therefore, a gain in electrical charge detection sensitivity. Secondly, piezoelectric elements and mechanical pieces are very
stable at low temperature which increases the overall stability of the
instrument and a very small mechanical drift.
In this chapter we are going to describe the development of this tool
and the first test we performed with it. The chapter is organized as
follows:
• In the first section we describe the general operating principles of
an AFM. This section is general for any AFM;
• In the second section we describe the construction of the microscope;
16
2.2. Principles of Atomic Force Microscopy
Figure 2.1: Scan electron microscopy image of a standard AFM tip
• In the third section, first tests are described. These tests are used
to calibrate the instrument and to characterise its performance.
2.2
Principles of Atomic Force Microscopy
In 1981, Binnig and Rohrer [19] reported the invention of the first near
field microscope: the scanning tunneling microscope (STM). For their
invention they were awarded the Nobel prize in 1986. In the press
release it was stated ‘that entirely new fields are opening up for the
study of the structure of matter’ [20]. Two decades later, the most
important scientific magazines confirm that this family of instruments
is at the heart of many of the most advanced research in solid state
physics. Part of the same family, the AFM invention came a few years
later [21]. It has the particularity of being able to probe any metallic or
non-metallic samples and to be a very little invasive technique. Here,
we describe the principles of operation of an AFM in the dynamic mode
and its fundamental limits.
2.2.1
Force Measurement
The heart of an AFM is a very sharpg tip located at the end of a
micro-cantilever. A scanning electron microscopy (SEM) micrograph
of an AFM tip is shown in Fig. 2.1. In the experimental setup the tip
is used as a probe while the cantilever is a spring that measures the
force between tip and surface.
In fact this idea is not original in itself as many instruments that
we utilise in our daily life take advantage of a spring to measure, for
example, the weight of a body (gravitational force). In this section,
we describe an analytical approach to measure forces using a spring.
Fig. 2.2 represents the conceptual idea of the system used to measure
17
Chapter 2. The Low Temperature Atomic Force Microscope
K
Fint
Figure 2.2: Scheme of the spring model. In this model the cantilever is
considered as a spring vibrating in a viscous fluid.
a force. In first approximation, we consider a one-dimensional massless
spring with an attached mass. The equation of motion is given by:
ω0 d
F
d2
z(t) +
z(t) + ω02 z(t) = ∗
2
dt
Q dt
m
(2.1)
where ω0 is the resonance frequency, Q is the quality factor of the
system associated with the internal damping and m∗ is the mass of the
object. The variable F describes the force acting on the object. Here
this force is separated in two contributions:
(2.2)
F = Fint + Fstimulation
where Fint is the force of interaction between the object and sample
and Fstimulation is an additional force introduced by an excitation signal.
In the dynamic mode we use an harmonic periodic signal to measure
forces and: Fstimulation = f cos(ωt). In this case the equation of the
spring model becomes:
d2
ω0 d
f
z(t) +
z(t) + ω02 z(t) = ∗ cos(ωt) +
2
dt
Q dt
m
fint
m∗
(2.3)
At free oscillation (i.e. Fint = 0) and for steady states the solution
is a harmonic oscillation with the same frequency as the stimulation
signal:
z(t) = Acos(ωt + φ)
18
(2.4)
2.2. Principles of Atomic Force Microscopy
0
a=A/A 0
Phase [Degree]
1.0
Du
0.0
0.97
1.00
1.03
-180
0.97
Du
1.00
1.03
u=f/f0
u=f/f0
Figure 2.3: Resonance curves under the influence of a force load: a) and
b) correspond respectively to the amplitude of oscillation (a) and phase (φ)
versus the reduced frequency (u). We plot both curves for a perturbed (solid
line) and unperturbed (dotted line) oscillator making use of expressions 2.5
and 2.6, respectively. In these images we used Q = 150 (typical quality
factor for an AFM in ambient conditions)
where:
A(ω) = s
Q2
A0
1−
φ(ω) =arctan Q
( ωω0 )2
ω
ω0
2
( ωω0 )
−1
2
(2.5)
+
( ωω0 )2
(2.6)
where A0 = (f Q)/k is the free amplitude at the resonance frequecy.
For Fint 6= 0, and assuming that the oscillator is perturbed by a small
force, the system can be described, in a first approximation, by the
first two terms of the Taylor expansion:
′
Fint (z) ≃ Fint (z0 ) + Fint
(z0 )(z − z0 )
(2.7)
In this situation, equation 2.3 can be rewritten as:
′
ω0 d
Fint
f
d2
2
z(t)
+
(ω
−
z(t)
+
)z(t) = ∗ cos(ωt) + C te
0
2
∗
dt
Q dt
m
m
where C te =
(2.8)
′ (z )z
Fint (z0 )+Fint
0 0
.
∗
m
19
Chapter 2. The Low Temperature Atomic Force Microscope
We conclude that, besides a constant term introducing a constant
deflection, the new resonance frequency (ωeff ) is:
2
ωeff
= ω02 −
′
Fint
(z0 )
m∗
(2.9)
and the shift of the resonance frequency is given by:
∆ω = ωeff − ω0 =
s
ω02 −
′
′
Fint
(z0 )
Fint
(z0 )
−
ω
≃
−ω
0
0
∗
m
2k
(2.10)
This shift in the resonance frequency of the oscillator is going to
be used to detect forces. Analyzing this expression we find directly
′
that for repulsive forces (Fint
(z0 ) < 0) the shift in frequency is positive
∆ω
′
(z0 ) > 0) the shift in frequency
( ω0 > 1) while for attractive forces (Fint
∆ω
is negative ( ω0 < 1). Moreover, we conclude that the dynamic mode
is sensitive to the derivative of the force rather than to the force itself.
This behaviour can be very useful if one wants to avoid constant forces.
Fig. 2.3 represents the amplitude (a(u)) and the phase (φ(u)) versus
the reduced frequency (u = ωω0 ). We plot both curves for Fint = 0 and
Fint 6= 0 making use of expressions 2.5 and 2.6. We used Q = 150
= 0.004 .
(typical quality factor for an AFM cantilever in air) and ∆ω
ω0
In the next section we will describe the topography principles using the spring model. Afterwards, we discuss the resolution and the
limitations that can be expected from this method.
2.2.2
Approach-Retract Curves and Topography Images
The question now is to know how we are going to build topographic
images making use of the shift in frequency when the tip probes surface
forces.
To better understand the whole process we start with the concept of
approach-retract curves. In these curves, the purpose is to investigate
the distance dependence of the data. We define approach-retract curves
as the recorded data (here amplitude and phase for the dynamic mode
of operation) while the tip slowly approaches to and retracts from the
surface. In this case we assume that the system is at each point in a
steady state.
From the previous linear model, we expect that both phase and
amplitude should decrease for a cantilever excited at the resonance frequency and if the tip only probes a positive force gradient. In contrast,
if the gradient is negative, we should see a decrease of the amplitude
and an increase of the phase. Seen from this perspective it becomes
20
2.2. Principles of Atomic Force Microscopy
32
c
Amplitude [nm]
a
10
d
20
0
40
DZ Piezo [nm]
60
Phase [degree]
b
-80
0
20
40
DZ Piezo [nm]
Figure 2.4: In a) and b) we show experimental approach-retract curves taken
at ambient conditions where a) is the amplitude and b) the phase. c) is
representating the principle of topographic images. As the tip scans over
the sample, a SiO2 surface, a feedback in amplitude is maintained to ensure
a constant distance between the tip and the surface. d) is a topographical
image taken at ambient conditions
clear that the phase is a sensor capable of distinguishing between contact and non-contact interactions[22].
In Fig. 2.4 we show, as an example, experimental approach-retract
curves over a SiO2 sample. In this case we observe a phase decrease
which means that the tip is always probing positive gradient forces.
To build topographic images the basic principle is to have a sensor
that guaranties that the tip-surface distance is constant. In the dynamic mode the cantilever is excited by a periodic signal giving the tip
an oscillating behaviour at the same frequency. During the scan the
amplitude of oscillation (smaller than the free oscillation amplitude) is
maintained constant by a feedback loop that serves the scan piezo to
approach or to retract from the mean position of the oscillating level.
In the meanwhile, the system records, at each x-y position of the sample, the z (vertical) displacement of the piezo needed to maintain a
constant amplitude. A two-dimensional image of the sample surface is
therefore built up. These constant amplitude images are, in the perspective of the spring model, constant force-gradient images. In fact, if
all electrical and mechanical properties are constant all over the sam21
Chapter 2. The Low Temperature Atomic Force Microscope
ple (i.e. if it is homogeneous and isotropic), then the constant gradient
is equivalent to constant distance, and consequently these images are
topographic images. An experimental approach-retract curve and a
topographic image are shown in Fig. 2.4.
2.2.3
Fundamental Limits
In this section we determine the physical limits of this tool. In our
experiments two factors are found to dominate the force sensitivity
and the image resolution: the thermal noise and the tip shape. Here
we evaluate the magnitude of these parameters.
Thermal Noise in a mechanical oscillator and Minimum Detectable
Force
In a mechanical oscillator there is always a natural oscillation provoked
by the finite temperature. It is caused by the brownian motion of the
molecules of gas surrounding the cantilever and the phonons of the cantilever. The fluctuation dissipation theorem quantifies the amplitude
and spectral distribution of this movement. Taking the equation of
movement of the lever (Eq. 2.1) and considering zb and Fb the Fourier
transform of z and F , respectively, the general solution in the Fourier
space is given by:
zb =
Fb
m∗



ω02
(ω02 −
−ω
2
ω2)
−
ω0 ω
Q
2
ω0 ω
Q
2 + i
(ω02 −
2
ω2)
−
ω0 ω
Q


2 
(2.11)
Now defining G the transfer function of the oscillator:
Fb
(2.12)
zb
the fluctuation dissipation theorem relates the spectrum density of the
tip motion, h|zb(ω)|2 i, to the imaginary part of G, Im[G], by:
G=
h|zb(ω)|2 i =
kB T
Im[G] =
πω
ω04
kB T
πkω0 Q (ω 2 −ω 2 )2 −( ω0 ω )2
0
(2.13)
Q
where kB is the Boltzman constant and T the temperature. Thus the
thermal noise excites the lever with a vibration noise N of:
N=
q
4πBh|zb(ω)|2 i
(2.14)
where B is the bandwidth of the detection system. The smallest detectable force derivative, δF/δz|mim , is the force gradient that produces
22
2.2. Principles of Atomic Force Microscopy
Figure 2.5: Schematic representation of the topographic convolution between
the tip and the surface. This phenomenon limits the topographic resolution.
In standard conditions, with a tip apex of 10 nm, the lateral resolution is
about 20 nm. The vertical resolution is nevertheless maintained and it is,
normally, better than 1 nm
a displacement equal to the thermal noise fluctuations. In this case we
have A0 ∆a = N . So, at the resonance frequency, one obtains [23]:
δF
δz
min
1
=
A0
s
27kkB T B
Qω0
(2.15)
For a typical AFM using typical values, i.e. k=1 N/m, B=300 Hz,
resonance frequency of 100 kHz, Q=300 and T =300 K the minimum
detectable force gradient is 2 × 10−5 N/m. Another remarkable conclusion
q is that decreasing the temperature to 4 K allows gaining a factor
of 300/4 ≃ 8.7 in terms of force resolution. So, at 4 He liquid temperature, the gradient resolution is 2.3 × 10−6 N/m.
Nevertheless, the thermal noise can be used as an elegant and precise way of determining the spring constant of the cantilever. An experimental thermal noise can be fitted by the Eq. 2.13. The spring
constant is the only free parameter.
Image Resolution
Here we make a remark concerning the resolution of an AFM image.
In the previous section we mentioned the force resolution. Nevertheless
this is not the only parameter. Another crucial parameter is the size
and shape of the tip.
In Fig. 2.5 we illustrate this principle. As the tip scans the surface there is a convolution between the tip and the morphology of the
23
Chapter 2. The Low Temperature Atomic Force Microscope
surface. The final result is always a smoothed image reflecting both
surface and tip shapes.
To obtain higher image resolution one must use either sharper tips
or shorter range forces [24]. Using commercial AFM silicon tips, most
of the experiments where atomic resolution is obtained are performed
at ultra high vacuum where surfaces are very clean [25]. In general,
at ambient conditions the resolution is limited by the tip apex. In
this case one obtains a lateral resolution of 10 to 20 nm and a vertical
resolution is sub-nanometric.
2.3
Conception of the Low Temperature Atomic
Force Microscope
The low temperature AFM was to a large extent designed and manufactured during the period of this thesis. The system was based on
a commercial [email protected] system (attoAFM I [3] [4]) to which an improved AFM head was included. Every piece was chosen carefully so
that the AFM could operate at low temperature and under a magnetic
field. In the next section the chosen options are going to be presented.
2.3.1
Requirements and Overview
This instrument demands three specific properties that are not usually
found in commercial AFMs:
• Low temperature operation;
• Ability of operating in the presence of a magnetic field;
• Versatility to host samples with several electrical contacts.
In response to these requirements, the AFM was designed to be
mounted in a cage-like structure formed by three bars. This structure
is inserted in a tube that is itself going to be inserted in a 4 He cryostat. The cryostat is equipped with a superconducting magnet that
can deliver a magnetic field up to 9 T at 4.2 K. In Fig. 2.6 we show the
project and a sketch of the concept of the microscope while in Fig. 2.7
we show photographs of the experimental setup.
The available space inside the cryostat is limited. In order to be
compact, the microscope is built around an optical fibre based interferometer. The cleaved single-mode fibre together with the tip lever
form a Fabry-Perot interferometer. This optical detection method was
24
2.3. Conception of the Low Temperature Atomic Force Microscope
shown to be able to detect vertical displacement of a cantilever with
high sensitivity [26] [27].
The microscope is made of non magnetic materials (mainly Al, Ti
and ceramic). These materials were carefully assembled in order to
compensate for thermal contractions. These two conditions permitted
the AFM head to work from 300 K to 4.2 K and under a magnetic
field.
During the construction of the microscope, the flexibility was another key point. The AFM is adapted to host samples that can be
simple metallic surfaces or complex devices that might have several
electronic contacts. In this respect special care was given to the cabling so that electrical contacts would be reliable without affecting the
mechanical stability.
Finally, the displacement of the sample under the tip is done via
x-y-z [email protected] inertial motors for the coarse approach [28] and a five
electrode scanner tube for a precise and smooth scanning when images
are being acquired. The inertial motors have a step-like behaviour permitting simultaneously a precise nano-positioning (resolution of 10 nm
at 4 K) and a long range travel in the centimeter range (7 mm) at low
temperatures and under magnetic field. A maximum bipolar voltage
of 120 V gives to the piezo tube a scan range of 30 µm in x-y directions
and a z-range of 3 µm at 300 K. At 4.2 K the scanner loses roughly an
order of magnitude making the lateral range limited to 3 µm and the
z-range to 800 nm.
The system is equipped with a commercial scanning probe high
voltage controller (Dulcinea Control System from [email protected] electronica
[29]). This electronics is composed of three modules: 1) lock-in amplifier used to demodulate the amplitude of vibration of the cantilever; 2)
a proportional-integral feedback system to build topographic images ;
3) and high voltage units to feed the piezo scanner. This electronic system permits the control of the piezoelectric scanner tube when images
are being obtained and the simultaneous acquisition of several signals
generated inside or outside the electronics.
2.3.2
Microscope Head and Optical Detection
In Fig. 2.8 we show a picture of the AFM head. The system was
designed in such a way that the tip can be adjusted in front of the edge
of the cleaved fibre to form a Fabry-Perot type cavity [3]. The light is
injected into this cavity through the fibre itself and the reflectance of
the cavity is collected by the same fibre. The tip is mounted on a stack
of two piezos: adjusting piezo and driving piezo.
The fibre is fixed in a holder by means of two teflon pieces and a
25
Chapter 2. The Low Temperature Atomic Force Microscope
AFM Head
He Cryostat
Single Mode
Fiber
Adjusting Piezo
Exitation Piezo
Tip Holder
Interferometer
AFM Head
Superconductiong
Magnet
Sample Holder
~20mm
Fabry-Perot
Sample Scanner
Inertial Motor
Figure 2.6: Schematic representation of the low temperature AFM. The
AFM is inserted inside a commercial 4 He cryostat and the optical detection
of the lever deflection is done via an optical single-mode fibre that forms an
optical cavity with the cantilever (see text). A detail of the project AFM
head is also shown.
26
2.3. Conception of the Low Temperature Atomic Force Microscope
a
Cryostat
Control and
acquisition
electronics
Optical
Detection
b
Mechanical isolating
springs
c
AFM Head
Sample Holder
Sample Scanner
Figure 2.7: Experimental setup. In a) we show an overview of the experiment. In b) and c) we show photographs of the microscope designed to be
inserted inside a cryostat.
27
Chapter 2. The Low Temperature Atomic Force Microscope
Tip Holder
Exciting Piezo
Fiber Holder
(with ceramics)
Adjustment
Piezo
Optical Fiber
Figure 2.8: Picture of the home made AFM head
small drop of glue next to the fibre edge. To anticipate for differential
thermal contractions the fibre holder is equipped with a ceramic with,
approximately, the same dimension and the same thermal coefficient
as the piezo stack.
The adjustment of the cavity is done manually. First the tip is adjusted in front of the fibre. Afterwards the fibre is approached towards
the tip. At the end of the process, the tip-fibre distance should be
around some tens of microns.
In Fig. 2.9 we show the theoretical curves of the light reflected by a
Fabry-Perot cavity as a function of the tip fibre distance. These curves
respect the classic Airy function:
1
IR
=1−
F ibre
I0
1 + F sin2 πD
λlaser
(2.16)
where I0 and IR are the light intensity injected and reflected from the
cavity, F is the coefficient of Finesse (which is directly linked with
the reflectivity of the mirrors), λlaser is the wavelength of the laser and
DF ibre the distance between fibre and cantilever. There are two relevant
points in these curves. Firstly, the amplitude of the fringes reduce very
strongly with decreasing F . Experimentally we observed that in our
experimental setup the coefficient of Finesse is around 0.03. Secondly,
two consecutive minima correspond to a displacement of λlaser /2. This
gives us a very accurate length reference for future calibration. Experimental curves will be shown in Fig. 2.10.
28
2.3. Conception of the Low Temperature Atomic Force Microscope
F=200
1
DFibre
I R /I 0
F=2
F=0.2
0
0
l/2
DFibre
l
3 l/2
Figure 2.9: Theoretical curves of the light reflected by a Fabry-Perot cavity
as a function of the mirrors distance. This curves were calculated for different
coefficient of Finesse F . Experimentally it was found that the tip together
with fibre form a Fabry-Perrot cavity with a F ≃ 0.05.
The fine adjustment is done making use of the adjusting piezo. The
tip is positioned at a point where the slope of Fabry-Perot curve is
maximal. At this point, if the tip performs a small displacement (i.e.
much smaller than λlaser /2), the fluctuations of light are proportional
to the amplitude of vibration of the lever. The proportionality factor
is the slope of the curve.
This effect permits the calibration of the cantilever deflection. If
one wants to do a quantitative measurement with the AFM this is a
very important point.
The optical setup is composed by two optical fibres that are sold
together, a laser diode and a photodiode detector. The coupled fibres
work as a fully fibreed beam splitter and permits the injection of light
inside the microscope and the separation of the reflected light. The
wavelength of the laser diode laser is 671.4 nm and the single-mode fibre
is adapted to this wavelength. The laser is equipped with a temperature
stabilized current source and a Faraday isolator to avoid re-injecting
light in the laser. Coherence length is announced by the constructor to
be around 50 µm.
29
Chapter 2. The Low Temperature Atomic Force Microscope
2.3.3
Cryostat and Mechanical Isolation
The experimental setup includes a commercial liquid 4 He cryostat. To
avoid a large consumption of He, the cryostat is composed of two concentric reservoirs. The external reservoir is filled with liquid nitrogen
and the central reservoir with liquid 4 He. The separation between the
two reservoirs and the separation between reservoirs and the external
environment are maintained in vacuum conditions. The tube, where
the microscope is inserted, is poured directly into 4 He.
The mechanical isolation from the environment is realized via four
big soft springs mounted under the cryostat. Furthermore, the exhaustion cryogenic tubes were chosen to be soft. Using this method, it was
found that the remaining mechanical noise came from the evaporation
cryogenic liquids.
To avoid a strong mechanical coupling between the cryogenic liquid
and the AFM head, the bars from the cage like structure were cut in
pieces of different lengths.
2.4
Testing
In the following section we describe the process of setting the microscope at low temperature and tests to ensure the good functioning of
the microscope. Afterwards we show typical results obtained with this
system and the calibration method we used.
2.4.1
Cooling of the Microscope
Before cooling, the system is pumped down to secondary vacuum to
evacuate any trace of water vapor. After this operation, 10 mbar of
He are introduced. This small pressure of He is used to thermalize the
microscope at low temperature. Then, the entire system is put to low
temperature. This procedure is done slowly to ensure no damage due to
thermal chocks. Typically, cooling the microscope takes three hours. In
the meanwhile, the good functioning of the AFM head is controlled by
the Fabry-Perot interference fringes. Experimentally we observe that,
because of thermal contractions, the fibre retracts from the tip some 50
micrometers. In Fig. 2.10 we show an interference fringe taken at low
temperature and a resonance curve of the cantilever. Using Eq. 2.16
we extract the coefficient of Finesse F = 0.03 ± 0.02. Furthermore, as
mentioned earlier, the amplitude of oscillation of the resonance curve,
on the right, is determined using the fact that two consecutive fringes
are separated by half of a wavelength of the laser. In this case the
conversion factor is 1.9 ± 0.2 V/nm.
30
2.4. Testing
7.0
Amplitude [nm]
Light Intensity [V]
6
4
2
-10
10
0
3.5
0.0
42130
42150
42170
Frequency [Hz]
Adjustment Piezo Bias [V]
Figure 2.10: On the left we show an experimental Fabry-Perot interference
pattern taken at 4.2 K. The fringes are used to calibrate the amplitude of
vibration of the cantilever. On the right, we show experimental resonance
curve of the AFM cantilever at 4.2 K. This curve is fitted by a lorenzian curve
which allows the determination of the resonance frequency (fR = 42.148 kHz)
and quality factor (Q = 9077)
.
a
c
b
Figure 2.11: (a) and (b) are approach retract curves, amplitude and phase,
respectively, done over a gold sample at 4 He. In (c) we show a topography
obtained in the same conditions
31
Chapter 2. The Low Temperature Atomic Force Microscope
Light Intensity [a.u.]
2.6
l/4 = 167.75 nm
1.8
linear region of detection
1.0
300
800
1300
Displacement [a.u.]
Figure 2.12: Approach curve used to calibrate the vertical displacement of
the scanning piezo.
The second step is to approach the sample towards the tip. For
that we use the inertial motors. In between each step we make one
approach attempt to ensure that we do not touch the sample. Once we
get close to the sample we obtain one approach-retract curve as shown
in Fig. 2.11 (a-b). Experimentally, this procedure is long due to the
non-reproducibility of the inertial motors. Furthermore, occasionally,
it was found that the inertial motor blocks. We unblock it by sweeping
the amplitude and the frequency of the electrical signal sent to the
motors. Typically, once the system is at low-temperature, it takes,
in average, one to two days to perform the approach. The very first
topographic image obtained with this microscope at 4.2 K is shown in
Fig. 2.11(c). The sample is a flat gold surface.
2.4.2
Calibration
In this section we describe the calibration of the system. This step is
crucial in the experiments that follow. We begin with the calibration
of the vertical displacement. Then, we describe the method used to
calibrate the x-y axes. At the end we show the procedure to calculate
the spring constant of a cantilever. This last step has to be done each
time a new tip is used.
Calibration of the scanner
The calibration of the vertical displacements is illustrated in Fig. 2.12.
In this image we record the static deflection of the lever as the tip
32
2.4. Testing
a
b
Figure 2.13: Topographic image of the grating used to calibrate the x and
y axes at ambient conditions (a) and at low temperature (b). The size this
square like holes is give by the manufacture
-6
2
2
(Amplitude) [nm /Hz]
6x10
-6
4.5x10
-6
3x10
42070
42120
42170
42220
Frequency [Hz]
Figure 2.14: Thermal noise of a cantilever at 4.2 K. From the fit it is possible
to determine the spring constant k = 4.37 N/m
33
Chapter 2. The Low Temperature Atomic Force Microscope
approaches a surface of silicon. For this purpose, the sample can be
considered as a hard material and so, once the tip touches the surface,
no more indentation is possible and it remains at a fixed point. After
touching the surface, the deflection of the cantilever coincide, therefore,
with the vertical displacement of the scanner. The interference fringes
observed in Fig. 2.12 are a direct measurement of the dilatation of the
piezo tube. Assuming that the piezo scanner has a linear response with
voltage, we are allowed to calibrate the vertical axes. We obtained a
calibration factor of 6.67±0.35 nm/V at 4.2 K.
The calibration of the x and y axes was done via a calibrated sample. In this sample we have several regions with a pattern of square
like holes with different dimensions. This dimensions are given by the
manufacturer. In Fig. 2.13 we show topographic images for both ambient conditions and liquid 4 He temperature. From these images we
obtain an horizontal plane piezoelectric factor of 278 nm/V at 300 K
and 28.7 nm/V at 4.2 K. In these values, we allow for an error bar of
15%. This error bar is due to the sample uncertainty and to the non
linearity of the piezo scanner.
Calculation of the spring constant of the cantilever
The thermal noise spectrum of a cantilever at 4.2 K is shown in Fig.
2.14. This spectrum is most important because it reflects the resolution of this instrument and it permits to calculate the rigidity of the
cantilever.
Observing this figure, the first conclusion is that the background
noise in our experimental setup (including shot-noise
of the laser, and
√
mechanical noise of the environment) is 1.8 pm/ Hz. At the resonance
frequency, even at low temperature, the noise level
√ is increased by the
thermal noise. The noise level value is 2.3 pm/ Hz.
As described in section 1.2.3, we can determine the spring constant
by fitting the experimental data with Eq. 2.13. From the experimental
thermal noise we calculate the spring constant as k= 4.37 N/m.
2.5
Summary and perspectives
In this chapter, we have described the working principles and the design
of a low temperature scanning probe microscope. We have seen the
first tests and the ability of the AFM to operate at low temperature.
Furthermore, we have calibrated both the in-plane and vertical axes
of the microscope scanner. This microscope has three advantages in
comparison with commercial AFMs. Firstly, it can operate inside a
34
2.5. Summary and perspectives
cryostat and in a presence of a magnetic field. Secondly, it permits the
determination of the amplitude of vibration of the cantilever. At last,
this AFM is versatile to host devices with several electronic contacts.
Nevertheless, we would like to number some possible improvements
that could be included while re-designing a new AFM. The improvements concern two aspects, the comfort while manipulating the AFM
and the noise level:
• Replacing the ceramics in the fibre holder (described in section 1.3.2)
by an inertial motor. This alteration will bring comfort and accuracy while adjusting the Fabry-Perot cavity. Furthermore, it
was also found that the fibre retracts some tens of µm while decreasing the temperature leading to a lost of sensitivity or, sometimes, to the complete loss of the optical signal from the cavity.
This improvement would completely compensate for this effect.
One should notice that there are, nevertheless, two inconveniences
when adopting this solution. Firstly, it will increase the fragility
of the AFM head. Secondly, six additional electrical contacts are
needed inside the cryostat.
• Protecting the driving piezo from undesired torque application.
As described previously, the AFM tip is glued in a tip holder
that it mounted on top of a pile of piezoelectric elements. We
found that successive positioning of the tip eventually breaks the
driving piezo. We suggest, e.g., a U-like structure inside which
the driving piezo would be glued. On the other hand, concerns
with the adjusting piezo do not seem to be justified as it had
shown to be must more resistant.
• Finally we would like to propose a damping system to the AFM
head. It was found that the springs mounted under the cryostat
were very efficient to isolate the AFM from any external sources of
mechanical noise. Nevertheless, a source of noise originating from
evaporation of cryogenic liquids remains. Solving this problem
does not seem to be an easy task, since mechanical springs become
rigid at low temperature. It is, however, and important issue if
the system is used to measure weak forces.
35
Chapter 2. The Low Temperature Atomic Force Microscope
36
Chapter 3
Electric Force Microscopy in
Parametric Amplification
Regime
Contents
3.1
3.2
Introduction . . . . . . . . . . . . . . . . . . . . .
Determination of the instability Domains of the
Cantilever Oscillation . . . . . . . . . . . . . . . .
3.2.1 Simple approach . . . . . . . . . . . . . . . . . .
3.2.2 Numerical approach . . . . . . . . . . . . . . . .
3.3 Cantilever Oscillation with ωel = 2 (ω0 + ∆ω) . .
3.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Experiments . . . . . . . . . . . . . . . . . . . .
3.3.3 Sensitivity and Thermomechanical Noise . . . . .
3.4 Parametric effects in Kelvin force microscopy .
3.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Experiments . . . . . . . . . . . . . . . . . . . .
3.5 Summary and Conclusions . . . . . . . . . . . . .
3.1
37
.
.
.
.
.
.
.
41
41
45
49
49
59
61
63
63
67
69
Introduction
In the beginning of this thesis we aimed at investigating and establishing a method to probe charges over a surface. For that purpose, we
have developed a theory, confirmed by experiments, that is capable of
increasing the sensitivity to detect electrostatic forces. The proposed
method uses a parametric amplification regime. Parametric amplification has already been observed in several mechanical systems such
37
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
as micro-actuators [30][31]. In an AFM configuration it was used once
to experimentally demonstrate thermo-mechanical noise squeezing [32].
Here, we further investigate this technique to detect small charge or potential differences between tip and surface. This model was developed,
to a large extent, by a member of our group, Thierry Ouisse, and the
results were published in [33] and [34].
Indeed, probing very small charges or electrostatic potential variations has become a key aspect in various fields of solid-state physics
[35][23]. In particular, the AFM has proven a powerful tool thanks to its
ability to measure small variations of local properties [36][37][38][39][40]
[41]. As a specialized technique, Electric Force Microscopy (EFM), a
derivate of dynamic force microscopy, is actively investigated for its
capacity to explore charge distributions in nanostructures[42]. Typically, EFM is performed as a dynamic method [36]. A sinusoidal electric signal is applied to the investigated substrate at a frequency ωel .
The electric force between the sample and the metallized tip results
in a modulation of the amplitude oscillations either at ωel or 2ωel , depending on whether the constant offset voltage or the alternating part
prevails, respectively.
Our experiments demonstrating the existence of such effects were
performed at ambient conditions. Nevertheless, we used the low temperature AFM, described in the previous chapter, for three reasons.
Firstly, our system is more versatile and ‘open ’ than typical commercial AFMs permitting full monitoring of the signals applied to the
system. Secondly, the optical detection allows amplitude calibration of
the tip vibration. Finally, these experiments served to test the stability
of the instrument.
An experimental example of the ability of the AFM to isolate electrostatic interaction from topography is shown in Fig. 3.1. In these
experiments electrostatic images are taken in two steps (each consisting of one trace and retrace) across each line. First topographic data is
taken, as explained in Chapter 1. Then, an alternative electrostatic signal (Vp ) is applied and the amplitude and the phase shifts are recorded
rising the tip up by a ‘lift height’. The second trace and retrace are
done with a constant separation between the tip and the local surface
topography, with the feedback turned off. This method has been shown
to be capable of distinguishing between forces of different ranges (e.g.
[5]). In fact, when topography is being recorded the dominant interactions are, in general, short-range repulsive interaction and van der
Waals forces. In contrast, for a tip at a distance around some tens of
nanometers from the surface and applying an alternative electric signal
these forces can be neglected. In this case, capacitive forces become
38
3.1. Introduction
a
~
Lift Height
b
Vp
V 00
c
d
Figure 3.1: Experimental topographic and electrical images taken at ambient conditions with our AFM. a) Scheme of the applied imaging mode where
the sample is scanned in a double passage to obtain independently the topography and the electrical measurements at a certain lift height and under
an applied voltage (Vp and V0 0). b) Topography of a thin film of ITO on the
top of which a layer of a mixture of polymer and molten salt was deposited.
(c) and( d) are electrical images (amplitude and phase) of the same region
at a lift height of 50nm with feedback off.
the most important contribution and so electrostatic images give direct description of the local capacitive interaction, i.e. the local charge
distribution. Fig. 3.1(b-d) shows experimental topography and electric
images (amplitude and phase), respectively, of a thin film of indium tin
oxide (ITO) covered by a layer of a mixture of conductive polymer and
molten salt. In this sample, the molten salt forms small domains that
are surrounded by the polymer. Inside the domains of molten salt the
anions and cations are free to orientate while the conjugated polymer
behaves as an insulator at small fields. Therefore, if a polarized tip
scans the surface we can clearly distinguish between the two materials.
These three images are very useful for demonstrating the ability of separating the van der Waals contribution, Fig. 3.1(b), from the capacitive
contribution, Fig. 3.1(c-d), and to differentiate the regions of polymer
from the domains of ionic salt. Indeed, the images of amplitude and
phase are, in first order, strongly correlated. Nevertheless, for a complete understanding of the tip-surface interaction both information are
needed [43]. These images were obtained at ambient conditions with
our AFM.
In this chapter we explore the frequencies of the electric signal ap39
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
plied to the tip in the context of a parametrical amplification model.
This approach covers as well the case of Kelvin probe force microscopy
(KFM), a widely used technique to measure local electrostatic properties of a sample.
To be able to use a parametric regime in EFM first requires a complete understanding of its various effects. Here, our aim is to provide
such an understanding. We present a fully analytical approach that is
crosschecked with numerical results. We also give some experimental
evidence of the model validity. Then, we propose to put it to good use
for improving the sensitivity of charge detection or surface potential
measurement. We note that in such a system any cantilever excitation
and thermo-mechanical noise are amplified in the same way, so that the
case of thermal noise has to be discussed separately. But parametric
amplification provides the ability to drastically reduce the impact of all
other noise sources. And even in the case of thermal noise, we suggest
a strategy to beat the conventional limit. These method makes use of
the fact that the information that we want to extract does not lie in the
raw piezo-electric excitation of the cantilever, which can be made much
larger than the thermal noise, but in the amplification gain, which is
tunable to a large extent.
The chapter is structured as follows:
• In section 2.2 we describe a numerical method to calculate the
instability domains of the cantilever oscillations, whatever is the
form of the electric signal applied to the substrate, and to any
degree of accuracy. The description of these instability domains
leads to the extraction of conditions that are well adapted for
charge or voltage detection.
• In section 2.3 we derive a fully analytical solution of the differential equation in the parametric regime with an electrical excitation
ωel around twice the natural frequency 2ω0 of the oscillator. We
follow the treatment described by Rugar and Grütter some years
ago [32]. Nevertheless we obtain contrasting conclusions. The
optimal setting of phase and voltage offset for voltage or charge
detection are deduced as a function of the main physical parameters of the system. This results are confirmed by experiments.
This analysis have permitted us to discuss the impact of thermal
noise on the electric measurements. We show that if thermomechanical noise is not the prevailing noise source, the predicted
sensitivity for charge detection can be increased by several orders of magnitude in comparison with the usual low frequency
detection scheme.
40
3.2. Determination of the instability Domains of the Cantilever Oscillation
• In section 2.4 we treat the case of ωel around ω0 , which also induces parametric amplification. This electric regime is commonly
known in literature as Kelvin force microscopy.
3.2
3.2.1
Determination of the instability Domains of the
Cantilever Oscillation
Simple approach
Before describing how we can get advantage of the parametric excitation to amplify the motion of an oscillator we start by determining
the conditions that make the dynamics of the cantilever entering into
an unstable regime. Here, our goal is to give simple arguments to
demonstrate the existence of such instabilities. To do so, we consider
a classical example that is treated in several textbooks, e.g. [44]: the
parametric pendulum.
A medieval example of such a system is the ‘Botafumerio’ in the
Cathedral of Santiago de Compostela, Spain. It consists on a mass of
56 Kg that is hung in a rope of 21 m long. This pendulum is swung at
a maximum angular amplitude of 80o and it is literally used to ‘spread
smoke’ inside the cathedral during special occasions. Impressively, this
rite is thought to be part of the liturgies for 700 years, i.e. several
centuries before the pendulum was studied. A detailed analysis of the
movement the ‘Botafumeiro’ can be found in [5].
Our analysis is much simpler. Our purpose is to demonstrate the
existence of unstable regimes. Indeed, when an oscillating system is
subjected to parametric amplification it may be driven into an unstable
state where non-linear contributions considerably influence the actual
motion.
In Fig. 3.2 we sketch the principle of a parametric pendulum. In
this case we consider a pendulum that is moved up and down at the
the point of the suspension. If the motion of this point is given by
z(t) = −Hcos(2ωt), the equation of motion takes the form:
d
1
d2
2
g
+
Hω
cos(2ωt)
sin(θ) = 0
θ
+
γ
θ
+
dt2
dt
l
(3.1)
where I denotes the length of the rope, g the acceleration of gravity, γ
the coefficient of friction and H and ω the amplitude and frequency of
the excitation, respectively. In this model we neglect the weight of the
rope and we consider the point mass approximation.
This system may be seen as the usual pendulum equation,(d2 /dt2 )θ+
(g/l)sin(θ) = 0 , plus two addition terms. The first one, γ(d/dt)θ, is
41
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
H
l
Fg
Figure 3.2: Scheme of the a parametric pendulum. We consider a case of
pendulum that is moved up and down at the the point of the suspension.
related to damping of the system due to the viscosity of the fluid surrounding the weight. The second term is the driving excitation which,
in this case, can be seen as a modulation of the gravitational acceleration g.
In this analysis we use the additional approximation of small angles
and so we are allowed to use the approximation: sinθ ∼ θ. In this
regime, and taking G(t) = g + Hω 2 cos(2ωt), we can rewrite Eq. 3.1 in
the form:
d2
d
G(t)
θ+γ θ+
θ=0
2
dt
dt
l
(3.2)
In a first approach let us consider that the system is not damped,
i.e. γ = 0. In this case we have:
G(t)
d2
θ
+
θ=0
dt2
l
(3.3)
This equation is formally identical
to the harmonic oscillator with
q
a resonance frequency of ω0 = g/l, except that G depends explicitly
on time. For this reason we call this oscillator a parametric oscillator.
The solutions of this kind of equations are not trivial. Nevertheless,
given that G(t) is periodic this differential equation takes the name of
Hill. Furthermore, since G(t) is also circular it can be further reduced
to the well known Mathieu equation.
42
3.2. Determination of the instability Domains of the Cantilever Oscillation
In this parametric regime the solutions of the differential Eq. 3.3
are of the form:
θ = eµt u(t)
(3.4)
where u(t) is a periodic function of time[45]. Instability domains, corresponding to µ > 0, can form in the frequency-amplitude plane of
G(t). Inside such instability domains, the oscillation amplitude is considerably increased with respect to the harmonic regime, and this phenomenon is at the origin of the amplification of parametric oscillators,
such as for the ‘Botafumeiro’ .
It is well known that for H = 0 we have the solution of the harmonic
oscillator: θ = θ0 cos(ωt + φ). Here, we argue that for a reason of
continuity, if H is close to 0, the solution of Eq. 3.3, θ, should be
approximately given by:
θ ∼ eµt cos(ωt + φ)
(3.5)
Substituting Eq. 3.5 in Eq. 3.3 and making zero the terms on ωt we
obtain a set of two coupled equations:
ω02 − ω 2 + µ2 +
H 2
ω cosφ − 2ωµsinφ = 0
l2
2ωµcosφ + ω02 − ω 2 + µ2 −
H 2
ω sinφ = 0
l2
(3.6)
(3.7)
These two linear homogeneous equations can be further reduced to:
µ4 + 2(ω02 + ω 2 )µ2 + (ω02 − ω 2 )2 −
H2 4
ω =0
l4
(3.8)
This 4th order equation in µ may be seen as a 2th order equation
in µ2 . Simple examination lets us conclude that this equation has
two real solutions, one negative and the other positive, corresponding
to the domains where the system is stable and unstable, respectively.
The solution of Eq. 3.3 is unstable if µ2 is positive. In this case, for
µ2 > 0, we obtain the following condition:
H
ω02
>2 2 −1
l
ω
(3.9)
Analogous analysis may be drawn for the case where the damping
factor, β, is not zero. Using the variable y = θexp(βt/2) we obtain:
!
d2 y
G(t) γ 2
+
−
y=0
dt2
l
4
(3.10)
43
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Instability
Domain
H/l
with
Damping
2γ/ω0
without
Damping
0
1
ω/ω
0
Figure 3.3: Instability domains, calculated using Eq. 3.9 and Eq. 3.13. We see
that in the frequency-amplitude plane the instability domains have tongue
shape around the resonance frequency ω0 .
which can be further cast into the form:
d2 y
+ α(t)y = 0
dt2
(3.11)
where α(t) = G(t)/l − γ 2 /4 is also periodic and circular. As defined
earlier Eq. 3.11 is also a Mathieu equation and the previous calculation can be repeated. We obtain, after some calculation, that in this
situation Eq. 3.9 is rewritten in the form:
2
ω02
H2
>
−1
l2 4
ω2
ω0
+
ω
2 H
ω0
2
(3.12)
Hence at the resonance frequency the system becomes unstable only
if:
H
2γ
>
l
ω0
(3.13)
The difference between the two situations can be better seen in Fig.
3.3. In this figure we draw the instability domains, calculated from Eq.
3.9 and Eq. 3.13. We see that in the frequency-amplitude plane the
instability domains have tongue shape around the resonance frequency
ω0 . It can be demonstrated that similar tongues for the frequencies
ω = 2ω0 /n, with n integer, may be found.
44
3.2. Determination of the instability Domains of the Cantilever Oscillation
Q
k
z
F
V 00
VP
Figure 3.4: Scheme of the our experimental apparatus in the point mass
approximation.
3.2.2
Numerical approach
As shown in the previous section, a parametric system may be driven
into unstable regime. Here, we will use a numerical approach to explore
the instability domains of the parametric system, so that we can find
conditions suitable for charge detection.
In Fig. 3.4 we illustrate our model to describe the dynamics of an
AFM probe under the influence of an electric field. As in Chapter 1,
we model the cantilever as a harmonic oscillator. Indeed, the equation
of motion of a cantilever is much more complex [46]. In general, the
equation of a flexural cantilever should be taken. In this case, the
dynamics of a cantilever is described by a fourth order equation and,
when working at high frequencies, the entire spectrum of frequencies
should be considered [47]. Nevertheless, the motion of the cantilever
may be substantially simplified if the investigated frequencies remain
much lower than the second flexural mode of the cantilever[48]. In the
case of a rectangular beam the ratio between the two first modes is
6.267 and in order to probe parametric phenomena we use frequencies
up to two times the fundamental resonance frequency of the cantilever.
In our model the oscillator is driven by both an external driving
force Fp (t) resulting from a piezoelectric bimorph and an electric force
Fel (t) exerted via the scanned substrate. The substrate is modelled
as a metal plane above which a fixed charge QF can be located, at a
given distance z1 from the metal plane. The equation of the cantilever
45
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
motion is:
ω0 d
Fpiezo (t) Fel (z, t)
d2
2
z(t)
+
z(t)
+
ω
z(t)
=
+
0
dt2
Q dt
m∗
m∗
(3.14)
with cantilever displacement z(t), spring constant k0 , mass m∗ and
quality factor Q. The resonant frequency ω0 is given by ω02 = k0 /m∗ .
The capacitive force Fel between tip and surface is given by [49]:
Fel =
1 δC
(V + ΦST + VC )2
2 δz
(3.15)
where ΦST denotes the work function difference between the metal
plane and the metallized cantilever and C is the capacitance between
the substrate and the cantilever. V (t) represents the voltage applied
to the substrate, while VC is an equivalent voltage which results from
the presence of the fixed charge. In the case of a plane-plane geometry,
Eq. 3.15 becomes:
ǫS
Fel =
2
V + ΦST +
z + z0
QF z1
ǫS
!2
(3.16)
where ǫ is the dielectric constant of the medium between the metal
plate and the tip. The average distance z0 accounts for the position of
the conducting plane set on a defined potential. A development of the
electrical force to first order in z turns Eq. 3.14 into:
!
d2 z
1 δFel
Fpieso (t) Fel (0, t)
ω0 dz
+ ω02 − ∗
=
+
z+
2
dt
m δz
Q dt
m∗
m∗
(3.17)
If the applied voltage V is an arbitrary periodic signal with pulsation
ω and period T , the homogeneous equation of motion has thus the form:
dz
d2 z
+ (ω02 − g(t))z + β
=0
2
dt
dt
(3.18)
with β = mω0 /Q. Due to the periodicity of V , g(t) is also a periodic
function of time. As in the last section we change the variable to
y = z.exp(βt/2) and we obtain:
d2 y
+ α(t)y = 0
dt2
(3.19)
where α(t) = ω02 − β 2 /4 − g(t). This equation is formally equivalent
to Eq. 3.3. If α(t) is a simple sine function, this equation is a Mathieu
equation.
46
3.2. Determination of the instability Domains of the Cantilever Oscillation
Figure 3.5: Instability domains in the electric signal frequency-pump voltage
plane, as calculated by applying the stability criterion to a sinusoidal electric signal. Parameters: S=2.25×10−14 m2 , m∗ = 10−11 kg, k=2.23 N/m,
Q=300, 150 fixed elementary charges located under the surface 25 nm and
distance tip surface of 50 nm
It is well known that Mathieu equation is formally similar to a
one-dimensional Schrödinger equation with a periodic sine potential
(replace t by x). Indeed, determining the instability domains of Eq.
3.19 is formally identical to determine the energy band structure of
an arbitrary one-dimensional periodic lattice. Moreover, it is of great
interest to solve Eq. 3.19 for any arbitrary signal, as electric losses in
the substrate changing the signal shape might have to be accounted
for, or the use of electric pulses might be favorable.
Fig. 3.5 is a map of the instability domains in the voltage-frequency
plane with the set of physical parameters listed in the figure caption.
In this calculation we have transposed a method originally proposed
by Lee and Kalotas in the case of the Schrödinger equation [50]. This
analysis has been done by T. Ouisse. Few more details may be found
in [33]. The black domains show the instability areas for a fixed charge
equal to 150 elementary charges. In contrast, the superimposed grey
domains are calculated for the same structure free of any fixed charge.
In both cases the electrical signal is sinusoidal. The grey domains
which are only due to the alternative part VP sin(ωel t) of the signal
are almost the same in both cases, and superimposed. The highest
frequency instability tongue due to VP is obtained for ωel = ω0 , since
it gives a force component oscillating at 2ω0 . In contrast, adding a
47
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.6: Instability domains in the electric signal frequency-pump voltage
plane, as calculated by applying the stability criterion to a sawtooth electric
signal with a linearly rising part and an abrupt falling part. Here VP is
defined as the amplitude of the sawtooth pulse and the base level is equal to
0 V. Parameters: S=2.25×10−14 m2 , m∗ =10−11 kg, k=2.23 N/m, Q=300,
no fixed charges.
constant offset through the introduction of the fixed charge gives rise
to a force component at ωel , and a new instability tongue thus appears
at ωel = 2ω0 . This is also quite clear in Fig. 3.6, for which the signal
is now a sawtooth periodic voltage. For low voltage values the two
first instability domains never exactly occur at ω0 and 2ω0 , but rather
take place at frequencies located below those values (see Figs. 3.5 and
3.6), because the resonance frequency of the cantilever is lowered by
the constant part of the first order term in z of the electrical force. As
detailed below, this is a favorable point for charge detection.
To enhance or lower drastically the natural oscillation amplitude of
the cantilever A0 through the action of a fixed charge or substrate voltage, the largest effect will clearly be obtained in the instability domains
described above, since in such a case the oscillations are controlled by
the non-linear terms in Eq. 3.14. But these oscillations can also become
spontaneous and difficult to control if the tip is positioned close to the
sample. Hence it is a priori preferable to stay out of the instability
domain, but sufficiently close to it so as to remain in a parametric amplification regime, which is amenable to analytical calculations. This is
for instance the case if one takes a sinusoidal signal with an electrical
pulsation ωel = 2ω0 . As illustrated by Fig. 3.5 for ωel = 2ω0 , as VP
increases from zero to upper values, one progressively gets closer to
48
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
Figure 3.7: Instability domains in the electric signal frequency-pump voltage
plane, as calculated by applying the stability criterion to a sawtooth electric
signal with a linearly rising part and an abrupt falling part. Here VP is
defined as the amplitude of the sawtooth pulse and the base level is equal to
0 V. Parameters: S=2.25×10−14 m2 , m∗ =10−11 kg, k=2.23 N/m, Q=300,
no fixed charges.
the instability domain, up to a voltage above which one begins to go
away from it. We thus expect an optimal amplification at a precise VP
value. A similar behaviour is obtained in the frequency-position plane,
as shown by Fig. 3.7. We will show that all quantities of interest can
be accurately approached by simple analytical expressions.
Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
3.3
3.3.1
Theory
In this section, we derive the gain in amplification obtained in the case
of a sinusoidal electrical excitation V (t) = V0 + VP sin(2(ω0 + ∆ω)t),
with ∆ω small against ω0 . Here, V0 includes either an offset voltage, a
work function difference, a fixed charge or a combination of the three
terms. We will follow below an analysis very similar in spirit to that
conducted by Rugar and Grütter in ref.[32]. As in [32], we introduce a
new complex variable:
dz
+ jω1∗ z
dt
dz
+ jω1 z
a∗ =
dt
a=
(3.20)
(3.21)
49
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
with
ω1 = ω0
s
j
1
+
1−
2
4Q
2Q
!
(3.22)
from which the following inverse relations can be obtained:
z=
a − a∗
j(ω1 + ω1∗ )
ω1 a − ω1∗ a∗
dz
=
dt
ω1 + ω1∗
(3.23)
(3.24)
Substituting Eq. 3.23 and Eq. 3.24 into Eq. 3.14 then yields:
kp (t) a − a∗
Fpiezo (t) Fel0 (t)
da
= jω1 a + j ∗
+
+
dt
m ω1 + ω1∗
m∗
m∗
(3.25)
where kp (t) is the modification of the spring constant induced by the
electrical force and Fel0 (t) is the zeroth order term in z of the electrical
force, depending on z0 . Eq. 3.25 is similar to what was already obtained
by Rugar and Grütter. However, we will not neglect any term in the
first order expression of the electrical force as made in [32]. We state
additionally that the piezoelectric stimulation is ensured at a frequency
half of the electrical excitation, so that parametric amplification acts
on the piezo-electric oscillation:
Fpiezo (t) = F0 sin ((ω0 + ∆ω)t + Φ)
(3.26)
Here we note an important difference with the treatment already
given in ref.[32]: the authors assume that the frequency ω0 involved
in their equations is not the natural resonance frequency of the free
cantilever, but rather the resonance frequency obtained with an offset
voltage V0 and pump voltage VP already applied between the substrate
and the cantilever. In other words, they implicitly include the constant part of kp (t) in the spring constant of the original system, and
neglect the variation of the quality factor, which is also a function of
V0 and VP . In their case, imposing ωel = 2ω0 means that the signal
lies exactly below the instability tongue, so that as the pump voltage
is increased, the gain continuously grows until spontaneous oscillations
appear. They also make the hypothesis that the pump voltage remains
negligible in front of the offset voltage, which is maintained constant
(as achieved in their experiment). Here, we seek a general solution
for an arbitrary frequency around 2ω0 , and arbitrary voltages. As a
50
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
consequence, our calculation results will also be different from that of
ref.[32]. Looking for a steady-state oscillation component of the form
a = a0 exp(j(ω0 + ∆ω)t), Eq. 3.25 leads to:
j(ω1 − ω0 − ∆ω)a0 −
×
"
!
ja
×
+ ω1∗ )
m∗ (ω1
#
Vp2
F0 j (Φ+ π2 )
V0 Vp ∗
2
a0 +
e
a0 −
(V0 +
=0
2
j
2m∗
(3.27)
1 δ2C
2 δz 2
(3.28)
where:
α=
We note that to first order in 1/Q, ω1∗ + ω1 ≃ 2ω1 and ω1∗ − ω1 ≃
jω0 /2Q [13], we define:
β1 =
ω0
2Q
(3.29)
V2
V02 + 2p
β2 = α
2m∗ ω0
(3.30)
β3 =
αV0 VP
2m∗ ω0
(3.31)
A0 =
F0 Q
m∗ ω02
(3.32)
where A0 is the free amplitude of the cantilever, already defined in
Chapter 1 (Eq. 2.5), i.e. when V0 = VP = 0 V. We find after some
calculation that a0 can be expressed as:
a0 =
−j
A0 ω02 (β1 + β3 )sinΦ − (β2 + ∆ω)cosΦ
−
2Q
β12 + (β2 + ∆ω)2 − β32
A0 ω02 ((β2 + ∆ω)sinΦ + (β1 − β3 )cosΦ)
2Q
β12 + (β2 + ∆ω)2 − β32
(3.33)
so that the oscillation amplitude A at ω0 + ∆ω is:
A0 ω0
A=
2Q
q
β12 + (β2 + ∆ω)2 + β32 − 2β3 (β1 cos(2Φ) + (β2 + ∆ω)sin(2Φ))
β12 + (β2 + ∆ω)2 − β32
51
(3.34)
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.8: Oscillation amplitude in the pump voltage VP offset voltage V0
plane, as calculated in Eq. 3.34. The iso-amplitude line values are given
in nm (from 0 to 4 nm with 0.2 nm steps). Parameters: ωel = 2ω0 ,
ωpiezo = ω0 , S=2.25×10−14 m2 , z0 =50 nm, k=2.23 N/m, m∗ =10−11 kg,
Q=300, Fpiezo =2×10−11 N, f0 =75.16 kHz. (a) Φ = 0 and (b) Φ = π/2.
(in Eq. 3.34, A is simply obtained from Eq. 3.33 by taking A =
|a0 |/ω0 ).
We should notice that Eq. 3.34 can be rewritten in the form:
(3.35)
A = GA0
where G is a gain that contains the information concerning the electric
field (and electric charges) probed by the tip. G is given by:
ω0
G=
2Q
q
β12 + (β2 + ∆ω)2 + β32 − 2β3 (β1 cos(2Φ) + (β2 + ∆ω)sin(2Φ))
β12 + (β2 + ∆ω)2 − β32
It can be easily verified that for V0 = VP = 0 V and ∆ω= 0 Hz G
becomes unitary. This is consistent with the fact that the parametric
excitation intervenes in the form a gain G.
Eqs. 3.34 and 3.36 provide the ability to calculate analytically almost any quantity of interest in the parametric amplification condition
ωel = 2(ω0 + ∆ω). Formula 3.34 is different from its counterpart derived in [13], since, as we will see below, A does not necessarily go to
infinity when increasing VP . Besides, for arbitrary VP and V0 values,
the maximum amplitude is in general not obtained for a phase shift
Φ = π/2, as in [13]. This can be seen in Figs. 3.8 (a) and (b), where
the amplitude of the oscillations has been plotted in the (V0 , VP ) plane
52
(3.36)
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
for Φ = 0 and π/2, respectively, and ∆ω = 0, the other parameters being analogous to that chosen in Fig. 3.5. In many areas the amplitude
with Φ = 0 is larger than with Φ = π/2.
From Eq. 3.34 it is straightforward to derive that if the condition:
2k
δ2C
>
2
δz
Q|V0 Vp |
(3.37)
is fulfilled, i.e. if the tip position gets close enough to the sample, then
the analytical gain goes to infinity if ∆ω is comprised in a frequency
interval given by the bottom and upper limits written below:
±
∆ω∞
"
1
VP2
2
=
−α
V
+
0
2m∗ ω0
2
!
±
s
α2 VP2 V02
k2
− 2
Q
#
(3.38)
For a sinusoidal electrical excitation, those two limits analytically
define the instability domains described in a more general way in the
previous section. In the case of a plane capacitor condition Eq. 3.38
becomes:
z<
4ǫSQ|V0 VP |
k
!1
3
(3.39)
An important quantity is the maximum negative frequency shift
∆ωmax which can be imposed without entering into the instability domain during an approach-retract curve. Finding the value of α which
+
minimizes ∆ω∞
and re-calculating the corresponding frequency shift
leads to:
∆ωmax
v
u ω0 u V0
= − t4
4Q
VP
2
+
VP
V0
2
(3.40)
Formulae 3.37, 3.38, 3.39 and 3.40 give the opportunity to precisely
select voltage, position and frequency conditions for which one can
get as close as desired to the instability domain. The most favorable
conditions for charge or potential detection are such that the gain is
maximized without entering into the domain characterized by spontaneous oscillations. The closer to the instability domain the system
is driven, the more sensitive is the method. A trade-off has thus to
be chosen between gain and stability. For instance, a quite safe but
somewhat restrictive measuring protocol might be to choose ∆ω = 0,
because the system never gets unstable and it even lends itself to a full
analytical treatment. One must seek the condition for which a small
53
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.9: Relative variation of the cantilever oscillation amplitude at ω0
versus the number of elementary charges added to a plane at z1 =25nm
from the substrate plane. Points: numerical integration of Eq. 3.14 and
solid line: analytical model. Calculation parameters: ωel = 2ω0 , ωpiezo =
ω0 , S=2.25×10−14 m2 , z0 =50 nm, VP = VP C =3.0567 V, Φ = −3π/8,
k=2.23 N/m, m∗ =10−11 kg, Q=300, Fpiezo =2×10−11 N, f0 =75.16 kHz.
variation in charge or potential (i.e. a change in V0 ) induces the largest
change in the oscillation amplitude, everything otherwise fixed. Hence
one must solve δA/δV0 = 0 or (δA/δV0 )/A = 0 if one is interested in
maximizing a relative variation. But before determining this condition,
we will first illustrate the precision of the analytical result through a
comparison of Eq. 3.34 with a numerical integration of Eq. 3.14.
Using a Runge-Kutta method of order 4, Eq. 3.14 was integrated
in a number of realistic situations with arbitrary initial conditions in z
and dz/dt, and the calculation was stopped when the oscillations were
getting sufficiently close to a steady-state regime (the integration time
was in the 50ms range). The relative variation of A versus the number
of elementary fixed charges added to the system is plotted in Fig. 3.9,
with ∆ω = 0, z0 =50 nm, z1 =25 nm, V0 =0 V and Φ = −3π/8 (as shown
later on this Φ value indeed gives the highest variation for V0 = 0 V).
It can be seen that the agreement with the numerical calculations is
excellent. Even for one elementary charge and a small surface capacitance, the variation in amplitude is around one per cent, and as the
charge grows the amplitude variation goes through a maximum, which
corresponds to the point closest to the instability domain. In Fig. 3.10
we simulated the variation of the amplitude with vertical displacement
z0 . Once again the agreement between numerical simulation and ana54
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
Figure 3.10: : Cantilever oscillation amplitude at ω0 versus displacement z0 ,
for various pump voltage values. Points: Fourier component at ω0 derived
from a numerical integration of Eq.(1) and solid lines: analytical model.
Parameters: ωel = 2ω0 , ωpiezo = ω0 , S=2.25×10−14 m2 , V0 =-1.2 V, Φ = 0,
k=2.23 N/m, m∗ =10−11 kg, Q=300, Fpiezo =2×10−11 N, f0 =75.16 kHz.
lytical result is excellent. One can appreciate the very specific shape of
the curve, which exhibits a maximum when the distance between the
tip and the substrate is such that one gets to the point closest to the
instability domain. As a matter of fact, all those maxima can be either
numerically or analytically calculated, as we will see below.
Fig. 3.11 (a), (b) give the variation of the sensitivity in the (V0 , VP )
plane, for the same parameters as in Fig. 3.8, and for two different phase
values. As illustrated by Fig. 3.11 there are always some extrema of
sensitivity in the plane; they will give a maximum relative variation of
amplitude if one makes a small change of offset voltage or fixed charge.
In particular, and as demonstrated below, taking Φ = −3π/8 gives rise
to one maximum on the line V0 = 0 V. Although in the general case
the determination of the extrema requires to solve an algebric equation
of order 7 in VP , one can choose simplifying assumptions, which give
the opportunity to find mere analytical expressions for all parameters.
Below we will focus on the case V0 = 0 V. Deriving Eq. 3.34 versus V0
and making V0 = 0 V in the resulting formula leads to the expression
δA
δV0
V0 =0
α
ω
2
F0 αVP − Q0 cos(2Φ) + 2m∗ ω0 VP sin(2Φ)
= − ∗2 2
3
2
4m ω0
ω0
α2
4 2
+
V
Q2
4m∗2 ω 2 P
(3.41)
0
55
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.11: Derivative of the oscillation amplitude divided by the amplitude
in the offset voltage V0 pump voltage VP plane, as calculated from Eq. 3.34.
The solid lines are for positive values and the dashed lines are for negative
values. Parameters: ωel = 2ω0 , ωpiezo = ω0 , S=2.25×10−14 m2 , z0 =50 nm,
k=2.23 N/m, m∗ =10−11 kg, Q=300, Fpiezo = 2 × 10−11 N, f0 =75.16 kHz.
(a) Φ=-3 π/8 and (b) Φ=0. Taking Φ=-3 π/8 gives rise to a maximum on
the line V0 =0 at VP = VP C2 (Eq. 3.46), whose position is indicated by a
black point.
Eq. 3.41 exhibits a maximum versus the phase if the relationship
below is obeyed:
VP2 = −
2k
tan(2Φ)
αQ
(3.42)
Here we note that Eq. 3.42 is also the condition for a maximum
sensitivity, i.e. a maximum of (δA/δV0 )/A . Substituting VP in Eq.
3.41 by the expression above, deriving versus Φ and finding the zeroes
of the resulting equation gives the conditions for which the phase and
the pump voltage maximize δA/δV0 at V0 = 0:
ΦC1 =
VP C1
π
π
+n
12
2
v
u
u
=t
2k
√
αQ 3
with n integer. δA/δV0 is then given by
s √
3
δA max
F0 Q 2 3α 3
=±
δV0 V0 =0
2 k
2
56
(3.43)
(3.44)
(3.45)
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
Figure 3.12: Relative variation of the oscillation amplitude at ω0 calculated
for an approach-retract curve without charge and with 10 added elementary
charges. The dashed line represents the maximum variation which might be
obtained by adjusting the pump voltage versus displacement z0 according to
Eq. 3.51. The two curves almost coincide when z0 is such that VP = VP C2 .
Parameters: ωel = 2ω0 , ωpiezo = ω0 , S = 2.25 × 10−14 m2 , z0 =50 nm,
z1 =25 nm VP =2,3,4,5 and 6 V, Φ = π/8, k=2.23 N/m, m∗ =10×−11 kg,
Q=300, Fpiezo =2×10−11 N, f0 =75.16 kHz.
Quite similar conditions are obtained for maximizing (δA/δV0 )/A:
ΦC2 =
π
π
+n
8
2
VP C1 =
1 δA
A δV0
s
(3.46)
2k
αQ
max
V0 =0
(3.47)
s
s
αQ
Q δ2C
=±
=±
k
2k δz 2
(3.48)
For a given z0 and V0 = 0, Eq. 3.48 gives the highest possible
sensitivity to a small charge or potential change which can be obtained
with ∆ω = 0. Any approach-retract curve with ∆ω = 0 will give a
sensitivity below Eq. 3.48. In the case of a plane capacitor model,
−3/2
this sensitivity varies as z0 . Now suppose that a value of VP has
been fixed and Φ = ΦC2 . There is indeed a given value of z for which
(δA/δV0 )/A is equal to Eq. 3.48. It is worth noticing that this is not the
point where a maximum sensitivity is reached in the approach-retract
57
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
curve (see Fig. 3.12 for an example). But interestingly enough, this
maximum of sensitivity is located close to that value. Easy calculations
show that with a fixed VP and Φ = ΦC2 , the maximum sensitivity of
the approach-retract curve is obtained for:
√
2(1 + 2)k
αC =
(3.49)
QVP2
and then is equal to:
1 δA
A δV0
max
V0 =0,Φ=ΦC2
1
= 1+ √
2
!
1
VP
(3.50)
For a plane capacitor, the tip position for which this maximum is
obtained is:
z0C
=
ǫSQVP2
√
2k(1 + 2)
!1
3
(3.51)
A possible way to measure the equivalent voltage difference between two points or resulting from charge injection into the structure
is to measure the maximum difference between the amplitude with and
without charge in an approach-retract curve. The equivalent voltage
difference ∆V0eq is simply given by:
∆V0eq
V0 =0,Φ= π8
VP
∆A
=
M ax
1.707
A
(3.52)
where ∆A/A is the relative oscillation amplitude variation between
the two approach-retract curves. Note that the lower is VP , the higher
will be the maximum sensitivity but it will then be obtained for a closer
distance between the tip and the sample, so that a trade-off must be
chosen in order not to get influenced by additional forces and effects
(see Fig. 3.12). The ultimate sensitivity of the method depends on this
choice.
Finally we will illustrate through Fig. 3.13 the gain brought by the
choice ωel = 2ω0 in comparison with the usual low frequency modulation of the cantilever oscillation which is obtained for ωel < ω0 . We take
ωel = 2π×10 kHz and all other parameters equal to the ones chosen in
Fig. 3.8, and we add a single fixed charge to the system at z1 =2.5 nm.
Fig. 3.13 shows the power frequency spectrum of the cantilever oscillations, as calculated from a numerical integration of the equation of
movement Eq. 3.14. As can be seen in Fig. 3.13 , there is an eight
orders of magnitude difference between the line at 10 kHz and the line
58
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
Figure 3.13: Power frequency spectrum of the cantilever oscillations as calculated from a numerical integration of Eq. 3.14 with the same parameters
as in Fig. 3.8, but ωel = 2π×10 kHz, VP = VP C2 =3.056 V (solid line) and
VP =10 V (dotted line).
at ω0 , the latter remaining almost unaffected by the electrical signal
at 10 kHz. This is to be compared with the one per cent increase in
amplitude of the line at ω0 in the case of parametric amplification (see
Fig. 3.8). Thus in this example we predict that the latter method gives
a six orders of magnitude improvement in sensitivity in comparison
with the former, if one takes into account all noise sources but thermal noise (to be treated in the last section). Increasing VP so as to
improve the sensitivity of the low frequency modulation does not substantially modify this ratio (case VP =10 V in Fig. 3.13). Besides, we
made our comparison in a very restrictive case (∆ω=0). If we choose
a frequency in between 2(ω0 + ∆ωmax ) and 2ω0 , the sensitivity of the
parametric method can still be greatly enhanced without suffering from
spontaneous oscillations.
3.3.2
Experiments
In this section we report data that are aimed at demonstrating without ambiguity the validity of the parametric amplification model, and
at identifying some critical parameters which must be carefully chosen
so as to put the method to good use (tip length, cantilever dimensions, etc.). The piezo-electric signal is picked up by an electronic circuit which exactly doubles the input signal frequency, with adjustable
phase, offset and level outputs.
59
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.14: Experimental approach-retract curves of a Cu plane in parametric amplification conditions, along with their fit with the analytical model.
The fitting parameters are: f0 = 57.402kHz, k=4N/m, Q=200, fel =2(f0 42Hz), fpiezo = f0 -42 Hz, Φ=0.904π. These parameters are either experimentally extracted (ω0 , k, Q) or chosen to be very close to the values given
by the tip supplier (cantilever dimensions).
Fig. 3.14 shows typical experimental approach-retract curves, the
sample being a flat copper surface, with an electric signal frequency
chosen to lie slightly below 2ω0 . As it can be seen in Fig. 3.14, application of Eq. 3.48 with parameters either experimentally extracted (ω0 ,
Q, k) or chosen very close to the parameter values given by the tip supplier (cantilever dimensions, etc.) allows us to obtain very reasonable
fits to the experimental curves, whatever is the offset voltage V0 . Here
the capacitance is modelled as resulting from the parallel combination
of a sphere+cone system (tip) and a plane capacitor (cantilever). For
the tip we use the analytical electric force model proposed by Hudlet
et al. [51]. It is worth noticing that a perfect fit cannot be obtained for
a number of reasons: for instance we do not take into account the 10o
cantilever angle, and the sphere+cone model is only an approximation
of the real geometry. Besides, the phase is also fitted because in practice there is a slight, constant de-phasing between the electric signal
applied to the piezo-electric bimorph which itself excites the cantilever,
and the resulting cantilever oscillation. Even with such an approximate
capacitance model it is quite remarkable that the whole set of curves
can be fit with only one set of parameters. Close to the sample, the
capacitance is dominated by the tip contribution, but far from it the
60
3.3. Cantilever Oscillation with ωel = 2 (ω0 + ∆ω)
Figure 3.15: Experimental variation of the oscillation amplitude as a function
of offset voltage, with the tip positioned at 0.33 µm from the sample surface.
Same parameters as in Fig. 3.14.
cantilever influence prevails. Fig. 3.15 gives the variation of the oscillation amplitude versus the offset voltage, now maintaining constant
the position of the tip with respect to the sample. It also illustrates
the nice agreement which can be obtained between the experiment and
model. It is worth noticing that if the parametric effect was ignored,
and just taking into account a ‘conventional’ excitation close to 2ω0 , i.e.
quite far from the resonance, the experimental approach-retract curves
should be almost entirely determined by the piezo-electric excitation
alone, and should remain almost unaffected by the sinusoidal electrical
force. This is obviously not the case in Fig. 3.14.
3.3.3
Sensitivity and Thermomechanical Noise
As already noted by Rugar and Grütter [32], if one is interested in
extracting data from the excitation which is parametrically amplified,
there is no gain to expect from the usual thermomechanical noise limit,
because both the useful excitation and the thermal fluctuations are
amplified in the same way. Indeed, this conclusion can be directly
extracted from Eq. 3.35 where the parametric excitation intervenes in
the form of a gain G. However, in this section we wish to point out that
the actual situation is somewhat more subtle, because in our case we
do not want to extract information from an arbitrarily small excitation
directly imposed upon the cantilever. We rather seek to obtain an
information which is contained in G, and not in the excitation to be
61
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
amplified. Thus the latter can be made much larger than the thermomechanical noise. In this section we briefly discuss this problem, in a
rather general way.
Suppose that one wants to determine the value of a physical parameter ν, which enters into the parametric force gradient, and not
into the primary excitation. If A0 is the amplitude of the piezo-electric
excitation before amplification, much larger than the thermal noise amplitude AB at ω0 + ∆ω, and if G is the parametric gain, a small change
∆ν in ν leads to an overall oscillation amplitude,
δG
A≃ G+
∆ν A0 + GAB
δν
(3.53)
The lowest detectable change in ν is obtained when GAB = A0 ∆νmin (δG/δν),
i.e. increase of the thermal noise is equal to the increase of the amplitude due to the variation of the physical parameter. We obtain therefore:
∆νmin =
GAB
δG
A
δν 0
(3.54)
It is clear from Eq. 3.54 that one can enhance the sensitivity just
by increasing the piezo-electric excitation, and that ultimately the sensitivity is controlled by the maximum excitation value which can be
safely chosen (for instance it cannot exceed the tip-sample distance),
and by finding the conditions which maximize the relative variation
of the gain with respect to the measured parameter, without entering into the spontaneous oscillation regime. To compare parametric
amplification with other methods therefore requires to compare expression Eq. 3.54 with the sensitivity given by the other technique,
and not to calculate the smallest, pre-amplified excitation that exceeds
the thermo-mechanical noise level. To compare parametric amplification with a conventional technique, one must evaluate the minimum
detectable amplitude when the force is now modulated at ω0 . The minimum detectable parameter value νmin is given by F (νmin )Q/k = AB ,
and this value must be compared to that given by Eq. 3.54. In fact, in
the regions where the amplification gain grows very rapidly, i.e. at the
limit of the instability domains, it is in general always possible to find
conditions for which Eq. 3.54 is indeed better than the conventional
limit.
62
3.4. Parametric effects in Kelvin force microscopy
3.4
3.4.1
Parametric effects in Kelvin force microscopy
Theory
In this section we complement the previous analysis for the case ω =
ω0 + ∆ω , i.e. when the cantilever is excited electrically close to the resonance frequency of the cantilever. This experimental AFM based technique is commonly known as the Kelvin force microscopy (KFM)[52][53].
In this technique, the average distance between the tip and sample z0
is usually maintained constant via a feedback control of the cantilever
oscillation imposed by a piezo-electric bimorph. The electric force Fel
results from the application of a sinusoidal electric signal V applied between the metallized tip and the substrate at a frequency ω = ω0 + ∆ω
close to the natural cantilever resonance ω0 . Although it is not ignored
that in addition to the ω component, there is also a 2ω force component
(Fel is proportional to V 2 ), its effects are usually neglected. However,
close to the sample, this 2ω component acts as a pump and induces
parametric amplification of the ω component. Depending on the operating point, the oscillation amplitude and sensitivity may substantially
depart from the expected ones.
Here, we use a analitical method analog to the one presented in the
previous section to describe the parametric effects in a KFM configuration, which is validated through experiments. Furthermore, we also
derive the best measurement conditions to enhance the measurement
sensitivity.
As in last section, the electric force can be described as Fel =
2
(V /2)δC/δz, where C is the tip/sample capacitance and V = V0 + VP
sin(ωt). V0 = V00 +∆Φ is the sum of the offset voltage V00 and the work
function difference ∆Φ between the tip and sample. In the conventional
modeling, the oscillation amplitude A due to Fel is proportional to both
V0 and VP , and ∆Φ is assessed by finding the V00 value which cancels the
oscillation. The sensitivity can be defined as the minimum voltage Vmin
for which A exceeds the noise amplitude AN at ω [52], A(Vmin ) > AN ,
and if Vmin is small this reduces to Vmin = AN /(δA/δV0 ). Hence the
best sensitivity is obtained by maximizing δA/δV0 , with V0 close to
zero, or by minimizing the whole quantity AN /(δA/δV0 ) if AN also
depends on the parameters to be tuned.
In a point-mass approximation and with a first-order development
of Fel the cantilever equation of motion is:
1 δFel (t)
ω0 d
d2
FP (t) Fel (0, t)
z(t)+
z(t)+ ω02 − ∗
+
(3.55)
z(t) =
2
dt
Q dt
m
δz
m∗
m∗
z(t), k, m∗ and Q are, as described in the last section, the cantilever
63
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.16: Oscillation amplitude in the V0 ,VP plane for a cantilever+tip
system and a metal plane, obtained from the parametric Eq. 3.56; capacitance modelled as in [13]. Parameters: cone length 15 µm, apex radius
100 nm, cone angle 5o , cantilever area 170×30 µm2 , tip sample distance
40 nm, k=2.5 N/m, Q=200, m∗ =10−11 kg. Iso-amplitude lines from 1 to
25 nm, with a 1 nm spacing.
displacement, the spring constant, the mass and the quality factor,
respectively. In Eq. 3.55 the periodic part of the electric force gradient
is usually neglected. However, when the tip approaches the sample
close enough this assumption is no longer justified, and this periodic
term makes the system parametric. Hence the 2ω electric component of
the force gradient may induce a parametric amplification of the forced
oscillation at ω (degenerate parametric mode [12]). Following the same
steps as in the previous section, i.e. keeping the time varying terms into
the force gradient and introducing normal variables, we obtain that the
oscillation amplitude at ω = ω0 + ∆ω is given by:
q
2
1 2QγV0 VP 1 + Q2 (α(V02 + VP2 /4)/k + 2∆ω/ω0 )
A=
(3.56)
k Q2 (αVP2 /4k)2 + (α(V02 + VP2 /2)/k + 2∆ω/ω0 )2 − 1
where 2α = δ 2 C/δz 2 and 2γ = δC/δz. From Eq. 3.56 the flat-band
voltage condition is still obtained by nullifying the oscillation with V0 .
For small V0 , VP and ω = ω0 , Eq. 3.56 reduces to the conventional
64
3.4. Parametric effects in Kelvin force microscopy
amplitude [52]:
Aconv =
−2QγV0 VP
k
(3.57)
However, A can also substantially differ from Aconv . In particular,
for small z0 and large VP , Eq. 3.56 gives:
A≃−
8V0 δC
δz
2
3VP δδzC2
(3.58)
which means that the amplitude can decrease with increasing VP or
when the tip gets closer to the sample (for a plane capacitor A is then
proportional to z0 ). This is in complete contrast with the conventional
model. Besides, from Eq. 3.56, A becomes infinite if VP > VP C =
(4k/Qα)( 1/2) and if ∆ω is comprised in the interval given by:
±
∆ω∞
=
−α(V02 + VP2 /2) ±
q
α2 VP4 /16 − k 2 /Q2
2m∗ ω0
(3.59)
As in the previous example (ωel = 2ω0 ) the actual motion will remain finite and it will be limited by the higher order terms neglected in
Eq. 3.55. But outside the instability domain defined by VP > VP C and
Eq. 3.59, Eq. 3.56 is a good approximation of the real amplitude. The
instability domain is not suited for KFM, because spontaneous oscillations can easily set in. But getting close to it enhances the parametric
effect and substantially modifies the oscillation amplitude. This is usually the case when the tip gets close to the sample, as required for a
local detection. Outside the instability domain, one can face two differ−
+
ent situations: either VP < VP C and ω0 −∆ω∞
< ω < ω0 +∆ω∞
, or ω is
located outside the frequency interval defined by Eq. 3.59. In the latter
case, due to the parametric effect, δA/δV0 does not grow indefinitely
with VP , but is maximized for a value which can be straightforwardly
derived from Eq. 3.56, and is given by a fourth degree polynomial equation. Although its analytical expression is complicate in the general
case, for V0 close to zero and ω = ω0 it reduces to:
2
VP OP T = 2 − 1 + √
3
1/4 k
qα
1/2
(3.60)
This is illustrated by Fig. 3.16, which shows the oscillation amplitude calculated from Eq. 3.56 in the V0 ,VP plane, for a realistic parameter set and ω = ω0 . Both A and δA/δV0 are higher on the line
VP = VP OP T . The parametric effect induces a profound departure from
65
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
the conventional analysis, for which the amplitude is proportional to
both VP and V0 , and the iso-amplitude lines are simple hyperbolae.
Hence, in contrast to a widespread belief, the sensitivity does not continuously improve with increasing VP , as suggested by the common
(and incorrect at high VP ) amplitude formula (Eq. 3.57). Indeed, without a careful parameter choice, the real amplitude can be either better
or worse than usually expected. For realistic systems, VP C is in the
range of a few volts. One can also choose to work in the potentially
unstable frequency domain, below ω0 . Then, as long as VP is smaller
than VP C , the frequency ωM which leads to a maximum amplitude AM
is derived by maximizing Eq. 3.56. A good approximation (to a few
per cent) is obtained by minimizing the denominator in Eq. 3.56. In
this case we obtain:
ωM = ω0 − α
VM =
Vp2 /2 + V02
2m∗ ω0
8γV0 VP (VP4 + VP4C )1/2
α(VP4C − VP4 )
(3.61)
(3.62)
This case is illustrated by Fig. 3.17, which depicts the variation
of the amplitude with frequency for the parametric and conventional
models, respectively. From the numerical solving of the differential
equation using the full electric force expression Fel = (V 2 /2)δC/δz
(points in Fig. 3.17), it is clear that only the parametric approach
correctly captures the cantilever motion.
From the above analysis we can now suggest optimised KFM measurement conditions. If the noise limit is not imposed by thermomechanical noise, the fluctuations are not affected by parametric amplification, and the best sensitivity is achieved with the parameters
which maximize δA/δV0 . Hence two KFM measurement strategies can
be considered: first, one can choose a conservative approach which
avoids the instability domain, with a given tip-sample distance and
ω close to ω0 . Then, one must apply the optimum pump voltage
VP OP T = 1.254(k/Qα)1/2 , which only depends on the capacitance coupling between the probe and the sample and the cantilever properties.
A second strategy is to work at frequencies below ω0 and belonging
to the potentially unstable range defined by Eq. 3.59. Then VP must
be kept below VP C , and it is preferable to work at ω = ωM . Such an
approach is more difficult to manage, for one must avoid to enter into
the instability domain, but the closer to it (i.e. the closest is VP to
VP C ), the better is the sensitivity. And this sensitivity can be made
much larger than the conventional one. However, if thermo-mechanical
66
3.4. Parametric effects in Kelvin force microscopy
Figure 3.17: Resonance curves estimated either from a numerical integration
of the full differential movement equation (points), the analytical parametric model (solid lines) or the conventional amplitude formula (dashed lines)
for a tip close to the sample; plane capacitor model. Q=300, k=2 N/m,
f0 =53.3 kHz, z0 =55 nm, capacitor area 0.1×0.1 µm2 .
fluctuations prevail, the noise amplitude at ω can be represented as
Ath = Ath1 sin(ωt) + Ath2 cos(ωt), where Ath1 and Ath2 are random amplitudes which vary slowly with respect to ω [32]. Ath1 has the same
phase as the electric signal and is amplified with the same parametric
gain G. Hence the signal/noise ratio is not expected to change with the
parametric effect, and remains the same as in a conventional analysis.
3.4.2
Experiments
To experimentally evidence this parametric effect we stabilized the tip
close to the sample, and we measured the resonance curves while varying VP . Typical results are shown in Fig. 3.18, obtained with an AFM
set-up already described in the last section, and a degenerate Si substrate covered with Au. Fig. 3.18 represents the resonance amplitude
AM versus VP , for z0 = 55 nm (i.e. the maximum in the experimental amplitude versus frequency curves). The variation is strongly nonlinear. Following section 2.3.2, we modeled the tip-sample with a sphere
+ cone geometry, in parallel with a plane capacitor corresponding to
the whole cantilever. The cantilever dimensions were that given by the
67
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
Figure 3.18: Experimental variation and fit to the resonance amplitude versus pump voltage, under ambient conditions. V0 =-80 mV, extracted parameters: k=7.3 N/m, Q=208, f0 =54.09 kHz; capacitance modelled as in [13]
with tip and cantilever dimensions as given by the tip supplier, fitted apex
radius r=0.34 µm.
tip supplier. k, Q and ω0 were experimentally extracted. The apex
size was adjusted (typically a few tens of nanometers). The analytical
expression 3.56 gives an excellent fit to the data. Here the offset from
the flat band condition is kept rather small (V0 ≃ -80 mV). Nevertheless, parametric amplification results into premature tapping around
VP =5.7 V. Without the parametric effect, tapping would have been
expected at VP ≃ 17.5 V. For distances larger than a few µm, the
amplitude is again linear with VP . Although such points are usually
not detailed in the literature, this might explain the usual practice of
operating with relatively small VP values, which here is justified by the
concern for avoiding premature tapping due to parametric amplification. Using small VP values reduces the sensitivity, but allows one to
avoid abrupt variations in amplitude. In contrast, close to the sample,
working with ω < ω0 avoids the amplitude vanishing which may result
from the parametric effect. From the above analysis, it is clear that for
optimizing KPFM measurements close to the sample parametric effects
must imperatively be taken into account, either for avoiding or using
them.
68
3.5. Summary and Conclusions
3.5
Summary and Conclusions
In this chapter we have shown that parametric amplification of a piezoelectric excitation by an electrical force may considerably enhance the
sensitivity of AFM tip oscillations to small changes in charge or offset
voltage between the tip and sample. Furthermore, in most interesting
cases the amplification gain is amenable to accurate analytical calculations, and thus this method could lead to practical applications in
EFM microscopy.
Nevertheless, here, we restricted ourselves to two Fourier components of the oscillation (ω0 and 2ω0 ), but in most cases the spectrum
is much richer and all other components may also include useful information, and can be analytically calculated.
We want to stress that even if parametric amplification with ωel =
2(ω0 + ∆ω) is not used, this phenomenon is not an academic problem,
but is very relevant in practice. The force resulting from an alternating electric signal intrinsically contains two frequencies, the value of
the latter being exactly twice that of the former. Hence in EFM with
sinusoidal electric excitation close to ω0 there is always parametric amplification, and its effects must be carefully analyzed, depending on the
measurement conditions.
This study made us conclude that, in contrast to a widespread belief, the ω component of the cantilever oscillation is not at all independent of the 2ω component of the electric force. We found that in KFM,
the parametric effect prevails and determines the oscillation amplitude
whenever the tip gets close to the sample. Furthermore, in the case of
ω = 2ω0 , we also showed that calculating the thermo-mechanical noise
limit is not as simple as with conventional techniques.
69
Chapter 3. Electric Force Microscopy in Parametric Amplification Regime
70
Chapter 4
Scanning Gate Microscopy of
Quantum Rings
Contents
4.1
Introduction
4.2
Electronic Interferences inside a Mesoscopic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.2.1
71
Aharonov-Bohm effect . . . . . . . . . . . . . . . . 74
4.3
Scanning Gate Microscopy . . . . . . . . . . . . .
77
4.4
Experiment . . . . . . . . . . . . . . . . . . . . . .
79
4.5
4.1
. . . . . . . . . . . . . . . . . . . . .
4.4.1
Quantum Ring . . . . . . . . . . . . . . . . . . . . 79
4.4.2
Experimental Setup . . . . . . . . . . . . . . . . . 82
Experimental results . . . . . . . . . . . . . . . .
87
4.5.1
SGM image Filtering . . . . . . . . . . . . . . . . . 88
4.5.2
Fringes analysis . . . . . . . . . . . . . . . . . . . . 91
4.5.3
Electrical AB effect in the SGM images . . . . . . 95
4.5.4
Imaging electronic wave function . . . . . . . . . . 99
Introduction
In the last two decades there has been a great interest in the electronic properties of the so-called mesoscopic systems [9][54]. This field
was motivated by the technological need for reducing the size of the
electronic circuits. This field tries to find answers to two questions:
• What are the novel physical properties when the size of electronic
circuits is reduced?
71
Chapter 4. Scanning Gate Microscopy of Quantum Rings
• How can we take advantage from this new physical properties to
build new electronic devices?
The mesoscopic physics deals, therefore, with systems that are much
larger than single atoms but whose electronic properties differ from
macroscopic samples. The word mesoscopic is derived from the greek
and denotes this characteristic. ‘Mesos’ means ‘middle’ or ‘intermediate’ while ‘skopien’ means ‘to look’. Indeed, new peculiar transport
properties appear at small dimensions, that are normally masked for
large samples. In this kind of devices signatures of electron interferences, such as Aharonov-Bohm (AB) interferences, that we are going
to see later, are observed.
Traditionally the study of these structures is done via macroscopic
measurements such as conductance measurements. While powerful
when coupled to statistical theories, this approach fails to provide a
spatially resolved image of the electronic properties.
The first attempt to obtain two dimensional images of electronic
systems used scanning tunneling microscopy (STM). With the STM
atomic resolution was achieved. Furthermore, only a few years after
the STM invention electron interferences could be visualized in real
space inside artificially confined surface structures. These structures
are known in the literature as "quantum corrals" [55].
However, since they rely on the measurement of a current between
a tip and the sample, STM techniques are useless when the system
of interest is buried under an insulating layer, as in two-dimensional
electron gases (2DEGs) confined in semiconductor heterostructures. To
circumvent this obstacle, a new method was developed: the Scanning
Gate Microscopy (SGM) [56][57][58][59].
In SGM experiments, instead of injecting a current, the tip is used
to perturb capacitively the sample. The principle is to use the tip as
a flying gate to perturb locally the potential experienced by electrons
within a device[60][61]. Changes in the electronic properties are measured indirectly by recording the conductance of the sample at every
perturbed point.
During this thesis we used a low temperature AFM, described in
chapter 1, to study the electronic transport inside a model system
widely known in literature: an open quantum ring (QR) [62][63]. These
experiments together with quantum mechanical simulations have permitted us to understand the mechanisms behind the formation of SGM
conductance images. We intend to develop an experimental technique
parallel to STM that would permit to probe the electron density in
devices buried a few nanometers under a surface.
This section is organized as follows:
72
4.2. Electronic Interferences inside a Mesoscopic Structure
• In section 3.2, we briefly introduce the readers to the electronic
transport inside a quantum ring and to the Aharonov-Bohm effect;
• In section 3.3, previous SGM experiments found in literature are
summarized and our experiments are situated in the SGM field;
• Samples and experimental setups are described in section 3.4;
• In section 3.5, experimental results are analyzed and compared
with quantum mechanical simulations.
At the end, conclusions are drawn and future perspectives are anticipated.
4.2
Electronic Interferences inside a Mesoscopic Structure
2DEGs are among the most popular mesoscopic systems and their
basic properties are very well documented in many reference books
(e.g.[64]). They form at heterointerfaces in layer semiconductor structures. Fig. 4.1 represents the energy diagram at the bottom of the conduction band in a typical heterostructure, as a function of the distance
from the surface. There we can notice a triangular-shaped well at the
interface between the two materials, originating from large differences
between their band gaps and the alignment of the Fermi levels in the
two materials. Electrons from nearby impurities are transferred to the
well where they form a thin conducting film (the 2DEG), leaving the
charged impurities behind them. With such a method extremely high
electron mobilities can be reached. Additionally, the confinement of
the electron systems to a very thin plane implies that electrons occupy
a discrete number of energy levels, or subbands. In the case sketched
in Fig. 4.1 only the lowest subband is populated.
In these kind of structures three fundamental characteristic lengths
rule the regime of the electron transport: the Fermi wavelength λF , the
electron mean free pass lm and the phase coherent length Lφ . Mathematical derivations and formal definitions of lm and Lφ can be found in
many mesoscopic textbooks (such as Datta [64]). Here, we will simply
explain intuitively the concepts behind these length-scales:
• λF is the wavelength of the electrons that contribute to the conduction inside the device. The energy of these electrons is the
Fermi energy and λF may beq
calculated from the electron density
(per unit of area), ns : λF = 2π/ns .
73
Chapter 4. Scanning Gate Microscopy of Quantum Rings
Energy
Location of the 2DEG
Material A
d-doping
Material A
Material B
Material B
+ + +
Fermi Level
2nd subband
1st subbband
Substrate
Depth
Figure 4.1: On the left we illustrate the profile of the heterostructure confining potential (sketched on the right) along the direction perpendicular to
the wafer surface. Electrons given by the doping layer are transfered to the
well forming, this way, a 2DEG.
• lm reflects the irregularities in crystals that lead to scattering of
the electron waves. It is defined as the average length that an
electron travels before loosing its initial momentum.
• Lφ is directly related to the wave nature of electrons. It characterizes the phase randomization. It is defined as the average length
over which an electron travels before loosing its initial phase. One
should notice that an electron can change its momentum without
loosing its phase coherence.
Lφ , λF and lm are, in general, distinct and can have very different values depending on the materials and the temperature. Advances
in processing technics permit, today, to produce (almost) arbitrary
small samples. In our experiments the size of the device was made
to be smaller than Lφ . In this case, the wave-like behaviour of electrons emerges and interference phenomena coming from different trajectories inside the device will eventually happen. Typical examples of
these inference phenomena are universal conductance fluctuations and
Aharonov-Bohm effect. In the next section we are going to describe
the AB effect, which was observed in our experimental data.
4.2.1
Aharonov-Bohm effect
The original figure of the experiment proposed by Aharonov and Bohm
is shown in Fig. 4.2 (a).
74
4.2. Electronic Interferences inside a Mesoscopic Structure
Their purpose was to demonstrate that ‘contrary to the conclusions
of classical mechanics, there exist effects of the potentials on charged
particles, even in region where fields (and therefore the forces on the
particles) vanish’ [65]. The experiment consists in an electron beam
that is split into two parts each going on opposite sides of a solenoid
and then recombine to interfere. In this article it was shown that the
magnetic potential provokes a phase shift of the electron wave function
going through the upper (φ1 ) and lower (φ2 ) arms. The phase shift
(∆φ) is given by:
∆φ = φ1 − φ2 = 2π
eZ
e
A dx = 2π Φ
h S
h
(4.1)
where Φ is the magnetic flux associated with the vector potential
A crossing the ring. As a consequence, two consecutive maxima are
obtained when Φ is increased by a factor of h/e which implies a periodical oscillation of the sample conductance as a function of the magnetic
field.
A particular case is obtained when the magnetic field is perpendicular to the ring surface. In this case Φ is simply the product between
the magnetic field (B) and the surface within the trajectories of the
electrons (S). The periodicity (TB ) is, therefore, given by:
TB =
h
eS
(4.2)
This means that if one knows the diameter of the trajectories it is
possible to predict the periodicity of the AB effect with respect to the
magnetic field.
In Fig. 4.2 (b), we show the first experimental observation of this
effect in a conductor [7]. In this case, the ring was fabricated in a
polycrystalline gold film of 38 nm thickness with a diameter of 764 nm.
Experimental results show clear periodic oscillations of h/e.
In the same article Aharonov and Bohm predicted a similar effect
using an electric potential, i.e. the electrical potential would also tune
the phase of the electron wave function inside the ring. This particular
phenomenon is known as the electric Aharonov-Bohm effect. In this
case the phase shift ∆φ provoked by an electric potential V is given by:
∆φ = 2π
eV t
h
(4.3)
where e is the elementary electric charge, h the Planck’s constant
and t the time during which electrons interact with the electric potential.
75
Chapter 4. Scanning Gate Microscopy of Quantum Rings
A
B
Figure 4.2: (A) is the original figure of the experiments proposed by
Aharonov and Bohm [65] to demonstrate electronic interferences in a split
coherent electron beam. In (B) we show the first experimental observation
made in an electronic system [7].
76
4.3. Scanning Gate Microscopy
The experimental verification in a metallic sample was done by
Washburn et. al. [8], where the electric field was created by means
of a lateral gate to the ring.
We underline that this effect can only be visible if electrons have a
wave-like behaviour. In other words, electrons must be in the coherent
regime. This effect is therefore a clear signature of the presence of
electrons in the coherent regime. Later in this chapter, in the part
devoted to the analysis of our experimental results, we use this effect
as a reference.
Nowadays this kind of phenomena still interests greatly the scientific
community. The objective is to build integrated electronic interferometers. Phase-controlled devices could be employed in complex coherent
circuits, where phase manipulation would be exploited for complex operations [66][67].
4.3
Scanning Gate Microscopy
The SGM was born as a variation of EFM. In the last decade, AFM
proved to be a very well adapted tool to study surface electric charges [22][13][14][68].
The key ingredient in this success is the localized electric field created
by the sharp tip. In most experiments, such as EFM, the AFM probe
is used to simultaneously perturb and measure the properties of the
sample surface.
Instead, in SGM experiments, the tip is used only to perturb the
sample [69]. It consists in mapping the conductance of the system as
the polarized tip scans at a constant distance above the 2DEG.
Prior to our experiment, two experiments have demonstrated the
ability to probe electronic transport in buried 2DEG.
The first experiment in this field was performed by Topinka et. al.
in Harvard [56][70][71][72]. In this article, the ability of SGM to image
coherent electron flow in a quantum point contact was demonstrated.
The central result is illustrated in Fig. 4.3. In this experiment, use
was made of a quantum point contact patterned from a GaAs/AlGaAs
heterostructure that was mounted in an AFM and cooled to 1.7 K.
The sample conductance versus the gate voltage on the quantum point
contact measurements show clear plateaus at multiples of 2 e2 /h (the
quantum of conductance). At each plateau, SGM measurements were
done. The images show the first three electron modes flowing through
the constriction. This observation was possible because the AFM tip
functions as a local interrupter. This regime, although very successful,
is limited to open 2DEG.
A further step in the study of mesoscopic devices was performed
77
Chapter 4. Scanning Gate Microscopy of Quantum Rings
Figure 4.3: SGM experiments on a quantum point contact reported in reference [56]. On the left, (A) schematic diagram of the experimental setup
and (B) conductance G versus gate voltage Vg showing clear plateaus at integer multiples of 2 e2 /h. On the right (A to C) are images of the electron
flow from the quantum point contact of three increasing widths corresponding to (A) the first conductance plateau, (B) the second plateau and (C)
the third plateau. These experimental results are compared with calculated
wavefunctions in (D to F).
by Crook et. al. [73]. In this case, the system was a quantum billiard
also patterned in a GaAs/AlGaAs substrate. To observe interferences
the AFM probe was used in a weak perturbation regime. Illustration
of the device together with the main results are shown in Fig. 4.4.
Fig. 4.4 (a) and (b) are SGM conductance images of the quantum
billiard in a magnetic of 42.0 and 39.1 mT, respectively. Image (c) is
a high-pass filtered image of (b). These experiments demonstrate that
the system is chaotic in the sense that a small change in the magnetic
field, for example, strongly modifies the conductance on an arbitrarily
small scale. Using geometrical considerations and a fractal analysis,
the authors linked the SGM images with scarred wave functions inside
a quantum billiard predicted by Akis et. al. in [74].
In our experiment, as we are going to see later, we studied smaller
78
4.4. Experiment
(a)
(b)
(c)
DG
(mS)
4
0
Figure 4.4: The experiment of Crook et.al. [73]. On the top, illustration
of the 1.4 by 2.9 µm2 quantum billiard. (a) and (b) are SGM conductance
images of the quantum billiard in magnetic fields of 42 and 39.1 mT. (c) is a
high-pass filtered image of (b) showing the scarred wave function inside the
quantum billiard.
samples (quantum rings) where this chaotic behaviour is no longer observed. The stability of QRs contributed to the understanding of the
evolution of the SGM images as the experimental parameters are swept.
Furthermore, the fact that is system is not chaotic permitted us to compare experimental results with simulations.
4.4
4.4.1
Experiment
Quantum Ring
During our experiments two rings were used, which we label R1 and
R2. The rings were patterned from an InGaAs/InAlAs heterostructure. This heterostructure was produced by molecular beam epitaxy
(MBE) in the IEMN ( ‘Institut d’Electronique de Microélectonique
et de Nanotechnologie’, Lille, France [75]) cleanrooms in the group of
Alain Cappy, Sylvain Bollaert and Xavier Wallart. A scheme of the
structure substrate is shown in Fig. 4.5. In this quantum structure,
79
Chapter 4. Scanning Gate Microscopy of Quantum Rings
doping 4.5 1012 cm-2
25 nm
Figure 4.5: Schematic layer structure of our heterostructure.
the 2DEG is formed at the interface between InAlAs and InGaAs. It
is located at 25 nm below the surface. At low temperature (4.2 K), the
electron density and mobility are 2 × 1016 m−2 and 10 m2 /Vs, respectively. The 2DEG is filled with electrons from a doping layer situated
a few nanometers above with a doping concentration of 4.5 × 1016 m−2 .
The rings were drawn in UCL ( ‘Université catholique de Louvain’,
Louvain-La-Neuve, Belgium[76]) using e-beam lithography followed by
wet etching. This procedure was done by B. Hackens (for more details
see [77]). Scanning electron microscopy images of rings R2 and R1 are
shown in Fig. 4.6 (a-b), respectively. The inner and outer diameters of
R1 are 200 and 600 nm while in R2 the inner and outer diameters are
240 and 614 nm.
In order to test the QRs we performed magneto-conductance measurements of each ring at 4.2 K. The corresponding curves are shown
under the micrographs in Fig. 4.6 (c-d). The conductances G of R1 and
R2 are around 5.9 and 2.6×2 e2 /h, respectively. Since the conductance
is larger than a quantum of conductance (2 e2 /h), which is the conductance for a single-mode electron flux propagating through a perfect
ballistic wire, it implies that several quantum modes are populated in
the device openings.
Superimposed on a broad background, periodic AB oscillations are
clearly visible (inset of Fig. 4.6 (c) and (d)). This effect is better shown
in the fast Fourier transform (FFT) of the magneto-conductance of R2
and R1 (displayed in Fig. 4.6 (e) and (f), respectively). This demonstrates that, at this temperature, the electronic transport through the
QRs has a coherent contribution, i.e. electrons have a wave-like behaviour permitting electrons to interfere. Furthermore, the FFT of
the magneto-conductances show several peaks where each peak corre80
4.4. Experiment
aa
b
597 nm
614 nm
200 nm
199 nm
232 nm
3.0
c
12.5
d
1.75
1.65
-0.45
-0.35
-0.25
2
G [2e /h]
2.5
2
G [2e /h]
1.70
2.0
9.0
9.58
9.43
9.28
-1.15
1.5
-2.5
2.5
B [T]
-1.35
3.0
AR1 B R1
20
2
2
22
-1.25
B [T]
f
2
AR2 B R2 CR2 DR2
FFT G [(2e /h) /B]
30
e
2
FFT G B [(2e /h) /B]
30
5.5
0.0
0
25
0
0
B
-1
30
-1
[T ]
35
40
150
0
30
0
0
B
-1
40
-1
[T ]
50
150
Figure 4.6: (a)-(b) show R2 and R1 electron micrographs, respectively. (c)
and (d) are the corresponding magneto-conductance curves of R2 and R1. In
the insets we evidence Aharonov-Bohm oscillations that are superimposed
on the broad background. (e)-(f) are the fast Fourier transforms of the
magneto-conductance shown in (c) and (d), respectively. The insets in (e)
and (f) evidence the existence of several peaks corresponding to several circular electron trajectories inside the QRs.
81
Chapter 4. Scanning Gate Microscopy of Quantum Rings
sponds to a well defined circular electron trajectory inside the QRs.
Concerning R1 we observe two peaks AR1 and BR1 with a corresponding periodicity of 26.1 mT and 25.0 mT. In the case of R2 we can
distinguish four peaks AR2 , BR2 , CR2 and DR2 with a periodicity of
33.3 mT, 32.2 mT, 30.8 mT and 28.3 mT, respectively. Using Eq. 4.2
and the peaks developed in the FFTs of rings R1 and R2 we determine
the corresponding radius of these trajectories (rR1 and rR2 , respectively). With this method we obtain that rR1 =[224 nm, 229 nm] and
rR2 =[199 nm, 202 nm, 207 nm, 216 nm] which closely matches the
interior of the QRs.
In the same magneto-conductance curves we can also see Shubnikovde Haas oscillations. They start developing at a magnetic field larger
than ∼ 2 T, i.e. as the cyclotron radius (calculated at 1µm for 2 T
and a carrier concentration of 2 × 1016 m−2 ) shrinks below the width
of the QRs arms and openings. In this case the electrons move along
the edges of the device and Shubnikov-de Haas oscillations develop.
4.4.2
Experimental Setup
Fig. 4.7 illustrates our imaging technique. A voltage Vtip is applied to
the tip, which scans along a plane parallel to the 2DEG at a typical
tip-2DEG distance (Dtip ) of 50 nm. Due to the capacitive tip-2DEG
coupling, a local perturbation is generated in the potential experienced
by electrons within the QR, which, in turn, alters their transmission
through the device. By recording the QR conductance as the tip is
scanned over the QR and its vicinity, we build a conductance map
G(x, y) that reflects changes in electron transmission.
While mounting the experimental setup special care was given to
the electronic noise. As shown in magneto-conductance curves in the
last section, the noise level should be better than 1/1000 so that coherent effects, such as AB oscillations, may be observed. In particular,
while cabling special attention was given to avoid mass-loops, which are
known to increase the noise level, while connecting the several instruments. Nevertheless this action was not sufficient, it was found that
the common mass of the institute was largely polluted by other instruments and other computers (of other experiments). As a consequence
most of the measurements presented in this section were performed
during quiet periods, i.e. weekends and nights.
The contact resistances to the QR are shortcut using a classical
four-contact measurement. To avoid low frequency electrical noise we
use an analog lock-in amplifier. We generate an alternative current
(I) in the lock-in, typically around 1 kHz, that is injected in the ring.
The resistive behaviour of the QR provokes a voltage drop (V ) at the
82
4.4. Experiment
I
V
Voltage
Source
AFM Tip
Sample
Magnet
Source
Scanner
Magnet
B
PreAmplifier
SPM Electronic
I
R
Lock-in
Amplifier
V
Figure 4.7: On the top we show a SEM micrograph of the quantum ring
(R1) and a representation of the experimental method. A scheme of the
experimental apparatus is shown below. A biased tip is used to perturb the
QR while the conductance is being measured using a 4-contact technique. In
the experiments the resistance of the QR is measured with a lock-in amplifier
and then transfered to the SPM electronics which also controls the sample
scan under the tip.
83
Chapter 4. Scanning Gate Microscopy of Quantum Rings
same frequency (typically some tens of microvolts). V is pre-amplified
and thereafter read by the lock-in. In order to avoid heating of the
electronic system by Joule effect, the current injected in the sample
should fulfill the condition eV << kB T . At 4.2 K we obtain that the
voltage drop should be V << 0.33 mV. With a typical resistance of
our samples of 5 kΩ the current is, in general, limited to 60 nA.
In these experiments we use, in the lock-in, a typical time constant
of 100 ms. This fact limits the scan frequency of the tip over the QR.
For an image of 128×128 points we use a scan frequency of 0.04 Hz (i.e.,
roughly 200 ms per point) and so each SGM image takes 53 minutes
to build.
In Fig. 4.8 we show images at different zooms of the sample mask
together with the QR. Fig. 4.8 (a) and (b) are micrographs of the
sample using an optical microscope and a SEM, respectively. In Fig. 4.8
(c) and (d) we show AFM topographical images of the QR at 300 K
and 4 K, respectively. In these images we can observe the four electrical
contacts to the QR. Furthermore, the sample is surrounded by a surface
of 200×200 µm2 of arrows guiding towards the QR. These arrows were
patterned together with the QR following the same procedure of e-beam
lithography followed by wet etching.
In these experiments we follow the following routine:
• The QR is mounted at ambient conditions and electrical contacts
to the sample are tested by means of measuring the resistance of
the QR.
• The QR is centered under the AFM tip. At ambient condition we
have optical access to the sample and we simply use a binocular
microscope to perform the coarse approach.
• Afterwards, the whole system is cooled using the procedure described in section 1.4.1.
• Once at 4.2 K, the optical access to the sample is no longer available. Therefore, to approach the tip towards the sample we use
the technique described in section 1.4. As stressed in the same
section, this procedure lasts for one to two days.
• We locate the ring and build a topography image. Typically, it
was found that, due to thermal contractions, the QR is misaligned
some hundreds of µm. The ring is re-centered using the arrows
drawn on the surface that guide towards the QR. This surface of
200×200 µm2 should be, in principle, larger then the misalignment. Nevertheless, due to an unexpected large misalignment, it
84
4.4. Experiment
Figure 4.8: Images of the sample mask together with the QR. (a) is a micrograph of the sample using an optical microscope. (b) is a SEM image
zooming over the rectangle marked in (a). In (c) and (d) we show further
zooms using AFM topographical images of the QR at 300 K and 4 K, respectively. In these images we can observe four electrical contacts to the QR
and a surface of 200×200 µm2 of arrows guiding towards the QR.
happened sometimes that the arrows region was lost during the
microscope cooling. In this case, we had to scan the surface of
the entire sample, leading to a loss of two to three days.
• We fit a plane on the topographic image and afterwards the tip is
left scanning the surface along this plane some tens of nanometers
(typically 50 nm) away from the surface. To verify the parallelism
between the scan plane and sample we let the tip scanning a few
nanometers away from the surface while recording the fluctuations
of amplitude. The parallelism is achieved when this image of the
amplitude fluctuations is symmetrical.
• The laser and the alternative excitation of the lever are turned
off.
• The QR ring is left stabilizing for one hour. This is necessary since
the semiconductor material is very photosensitive and is disturbed
by the laser beam injected into the cavity for the topography
acquisition.
This strategy takes in average one week to be performed. After this
procedure, we are ready to collect images of the QR conductance. In
85
Chapter 4. Scanning Gate Microscopy of Quantum Rings
Driving Piezo
Fibre
Holder
AFM cantilever
Electrical
Contacts
Copper
Piece
Sample
Sample
Holder
Termometer
Heater
Piezo
Scanner
Copper
Piece
Heater
Sample Holder
Termometer
Sample
Electrical Contacts
Figure 4.9: On the top, photograph of the experimental setup taken just
before being cooled. In this case the system was also equipped with a thermometer and heater. On the bottom, scheme of the heating process. All
three elements (heater, thermometer and sample) are glued to the same copper piece to guaranty an homogeneous temperature between them.
the next sections we are going to show and analyze the experimental
results.
Heater and Thermometer
During these experiments we used a system to heat the sample. Indeed, many electronic properties are very sensitive to temperature variations [78]. By following their temperature characteristic many effects
can be compared and identified.
With this in view we conceived a system adapted to enter the AFM
that would simultaneously permit to heat and to measure the temperature of the sample. A photograph of the experimental setup taken
just before its cooling is shown in Fig. 4.9. A schematic representation
of the setup is shown at the bottom of Fig. 4.9. This system is com86
4.5. Experimental results
posed of three elements, all glued to the same copper piece to guaranty
an homogeneous temperature. The copper piece is thereafter glued to
the sample holder. The heater is a resistive wire that dissipates, by
Joule effect, energy to the sample. A good thermal coupling between
the sample and the thermometer is guarantied by gluing both of them
with silver paint to the copper piece. The thermometer was chosen to
have a high sensitivity in a temperature range of 1 K to 100 K. For this
purpose we equipped our experimental setup with a CERNOXT M temperature sensor made of a thin-film resistance of zirconium oxy-nitride
[79].
4.5
Experimental results
As already explained, in our experiments, conductance maps are recorded
while scanning the biased tip of a cryogenic AFM above the quantum
ring and its vicinity. In this section we analyze the well developed
patterns observed in these images. These results are presented in the
following way:
• In section 3.5.1, we analyze the raw SGM data and we propose a
high-pass filter adapted to isolate coherent effects.
• Afterwards, in section 3.5.2, we distinguish different types of
fringes on the filtered images (concentric outside the ring region
and radial inside the ring region). To evidence this different behaviour we sweep the tip voltage and a DC current injected in the
QR.
• In section 3.5.3., we observe that the AFM probe induces a phase
shift on the electrons crossing the ring. This effect is interpreted
in terms of an electric Aharonov-Bohm effect produced between
the tip and the rings arms.
• In section 3.5.4 we include quantum mechanical simulations to
demonstrate that the images obtained by SGM have a strong relation with the electronic probability density inside the QR. This
has permitted us to investigate the origin of concentric fringes.
These results are compared with the experimental data shown
previously.
At the end we summarize the achievements obtained in this experiment. The results presented in this section have recently been
published in two publications [80] and [81].
87
Chapter 4. Scanning Gate Microscopy of Quantum Rings
4.5.1
SGM image Filtering
A typical example of a conductance map, measured in R1, is shown in
Fig. 4.10. These G maps reveal a rich pattern of conductance fringes
superimposed on a broad and slowly varying background.
The first contribution to the map is a broad background structure
extending beyond the limits of the QR. The associated conductance
scale is typically 2 e2 /h, i.e. almost an order of magnitude larger than
the amplitude of the AB oscillations of Fig. 4.6 (c-d), indicating that
the background does not originate from coherent effects. Moreover,
we observe experimentally that its overall shape is strongly affected
by successive illuminations of the sample which are known to affect
the configuration of ionized impurities, but remains insensitive to the
magnetic field B. This indicates that this background is not due to
coherent effects which are known to be very B-sensitive, but related to
a global shift of the electric potential in the whole quantum ring as the
tip approaches the quantum ring.
A closer look to the conductance map reveals that the broad background is decorated by a more complex pattern of smaller-scale features, particularly visible in the central part of the image. We isolate
these features by applying a high-pass filter, shown in Fig. 4.10 (f), on
the raw conductance map. Fig. 4.10 (g) is obtained by applying such
filtering on Fig. 4.10 (a).
Fig. 4.10 describes the successive steps followed to choose the cutoff frequency (fcut ) of the filter that we apply to our raw conductance
maps. First, as we change Vtip , we notice that the background of the
conductance map remains essentially unchanged, while the position of
the small-range features is significantly affected. Therefore, after averaging over a sufficient number of conductance maps acquired with
different Vtip (in this case seven images), the small-range features are
averaged away, leaving only the background (Fig. 4.10 (b)). Thereafter, we compare the fast Fourier transform (FFT) of the averaged
image (Fig. 4.10 (d)) with a typical FFT of a raw conductance map
(Fig. 4.10 (c)). To better distinguish the two frequency ranges, in
Fig. 4.10 (e) we show the folding of Fig. 4.10 (c) and (d) in one axe.
In this figure we can clearly distinguish between two frequency ranges.
The low-frequency range, where the two FFTs are essentially identical,
corresponds to the broad background structure in the real-space conductance map. The high-frequency range, where the spectral content
is much reduced in the averaged FFT, is related to the small-range
features that we want to isolate. The cut-off frequency fcut is chosen
at the transition between both ranges, i.e. fcut = 4 µm−1 . The image
of the filter, in the Fourier space, is shown in Fig. 4.10 (f).
88
4.5. Experimental results
a
b
2
G [2e /h]
10
0
10 0
10
d
50
averaged data
raw data
-1
Frequency [mm ]
-10
fcut
0
0
0
10
5
20
10
15
-1
20
Frequency [mm ]
Fourier power [a.u.]
10
0
g
f
-10
-1
Frequency [mm ]
10
8.32
7.65
e
Fourier power [a.u.]
c
2
G [2e /h]
8.32
7.65
0
1
2
D G [2e /h]
-0.2
0.0
0.2
Figure 4.10: Image filtering technique. (a) conductance of the QR as a
function of the tip position with Vtip =0.3V, tip-2DEG distance of 50nm and
B=2T. (b) is the average of seven SGM images taken for different Vtip . (c)
and (d) are the fast Fourier transform (FFT) of (a) and (b), respectively.
In (e) we compare the FFTs by folding the FFT of the averaged image (d)
and the FFT of a single image (c) in a plane. The cut-off frequency (fcut ) is
chosen to isolate the high-frequency fringes superimposed to the background.
From (a), using the high-pass filter with fcut =4 µm−1 shown in (f), we obtain
the image (g).
89
Chapter 4. Scanning Gate Microscopy of Quantum Rings
d
0.06
28.0 K
DG [2e 2/h]
a
T = 4.4 K
T = 4.4 K
3.8 K
0.00
-0.2
g
-0.1
0.1
0.0
0.2
B [T]
e
b
0.01
f
,raw
c
2
d G SGM [2e /h]
T = 13.8 K
2
d GB [2e /h]
T = 13.8 K
0.05
3
T = 26.1 K
T = 26.1 K
3
2.50
G [2e 2/h]
2.70
0.06
-0.06
DG [2e 2/h]
40
0.001
10
h
0.005
40
T [K]
Figure 4.11: Comparison between the temperature dependence of the nonfiltered SGM maps (a-c), filtered SG maps (d-f) and AB interference fringes
(g). (g) is ∆G vs B at 3.8 K and 28.0 K in R2. (d-f) are ∆G(x, y) filtered
maps of raw maps in (a-c), respectively, measured on sample R2 at Vtip = 0V,
Dtip = 50 nm, B = 0 T and T = 4.4, 13.8 and 26.1 K, respectively. (h) Red
circles: standard deviation of the magneto-conductance curves δGB vs T
(left axis); blue triangles : standard deviation over the ring region of filtered
SGM images δGSGM vs T (right axis). The black line is a guide to the eye.
The inset shows the standard deviation over the ring region of original SGM
maps δGSGM raw vs T .
90
4.5. Experimental results
Another aspect of filtering the experimental images is to avoid artifacts created by the borders of the image. In order to avoid these
effects before filtering the raw image, the raw image is extended by 256
points per axe. This extension is done by taking the value of the last
point of each line of the image. After applying the filter to the FFT of
the raw data we perform the inverse FFT and, at the end, we zoom up
the center of the image (128 per 128 points).
An additional argument for filtering the images is shown in Fig. 4.11.
Fig. 4.11 (d-f) show high-pass filtered ∆G maps of Fig. 4.11 (a-c),
respectively, measured on R2 between T = 4.4 and 26.1 K.
In Fig. 4.11(d-f), we note a decay of the ∆G fringe amplitude with
increasing T , but the fringes pattern remains essentially unchanged.
We define δGSGM , the standard deviations of the filtered conductance
maps, calculated on a 400 nm diameter circle centered on the QR, and
δGB , the standard deviation of AB oscillations in G vs B. We can compare the T -dependences of SGM images and AB effect (Fig. 4.11(h)).
δGB and δGSGM clearly follow the same T -dependence: a strong decay above T ∼ 10 K, and a saturation at lower T , consistent with
the intrinsic saturation of τφ previously reported in similar confined
systems [24]. This suggests that the of the AB effect and the central
pattern of fringes in ∆G maps are intimately related and find a common origin in electron wavefunction interferences. This will be further
discussed later on.
Here, we want to stress that similar examination was performed
with the raw data. In the inset of Fig. 4.11(h) we show the evolution
of amplitude of the fringes from the raw data δGSGM raw with temperature. In contrast with the filtered images, this curve follows a very
distinct temperature evolution from δGB and δGSGM indicating that
the background does not share the same coherent origin. Therefore,
the filtering technique is very powerful in extracting coherent effects
from raw data.
In the remaining of this chapter, we focus on high-pass filtered images, and try to uncover the origin of the oscillations visible in the
data.
4.5.2
Fringes analysis
In Fig. 4.12 (a-c) we show a sequence of SGM conductance maps for a
tip-surface distance Dtip of 50 nm, 70 nm and 90 nm. In this sequence
we observe that the amplitude of the fringes decreases as Dtip increases.
Nevertheless, the fringes are maintained in all the three images. This
effect can be better observed in Fig. 4.12 (d-e). In Fig. 4.12 (d) we
plot the circular profile under the ring region of images shown in Fig.
91
Chapter 4. Scanning Gate Microscopy of Quantum Rings
DG[2e 2/h] -0.065
a
b
c
0.065
concentic
fringes
radial
fringes
0.04
e
d
2
90 nm
70 nm
b
2
c
a
0.00
-0.1
0
Circular Distance [nm]
300
dG Profile [2e /h]
Profile D G [2e /h]
50 nm
0.1
0.00
-1
0.02
1/Dtip [nm ]
Figure 4.12: (a-c) sequence of SGM conductance map for a tip-surface distance Dtip of 50 nm, 70 nm and 90 nm. (d) profiles of images (a-c) over a
circle inside the ring region. In (e) δGProfile denotes the standard deviation
of each profile vs Dtip . The line correspond to a distance dependence of
1/Dtip . In this set of images we used Vtip = 0.3 V and B = 1 T
4.12 (a-c). Afterwards, the standard deviation is calculated for each
circular profile and plotted versus Dtip in Fig. 4.12 (e). This result
demonstrates the stability of the SGM images over a QR, i.e. there is no
sign of chaotic behaviour. Furthermore, the typical spatial periodicity
of the oscillations (∼ 100 nm) is much larger than the electron Fermi
wavelength in our sample (λF ∼18 nm). These two evidences are in
contrast with previous experimental studies of Crook et. al., shown
in section 3.3, and tells us immediately that the ‘standing electron
wave’ pattern observed is not the relevant mechanism to explain our
observations.
As shown in Fig. 4.12 (c) the fringe pattern depends on the scanned
region. While fringes are predominantly radial when the tip is located
directly above the ring, i.e. δG fringes emerge from the centre of the
QR towards its perimeter, they become concentric when the tip moves
away from the QR with a larger amplitude on the left side of the QR. It
is worth mentioning that this left-right imbalance is observed whatever
the direction of the magnetic field or the probe current. Therefore,
it is related to an asymmetry of the QR or tip shape, and is not a
signature of the Lorentz force which could also lead to an imbalance of
92
4.5. Experimental results
b
a
I
II
III
I DC = +0.8 mA
0.02
dG [2e /h]
c
2
I
II
I DC = 0.0 mA
d
III
0.01
-1
I DC = -0.8 mA
0
1
0.06 0.00 -0.06
I DC [mA]
2
D G [2e /h]
Figure 4.13: Evidence for a different behaviour of the standard deviation
of radial and concentric fringes. (a) δG (standard deviation of ∆G) as a
function of the probe current IDC , calculated in different areas (I-III) of the
filtered conductance maps, indicated in (b). (b-c) filtered conductance maps
recorded at B=5.95 T, for different probe currents : +0.8 µA (b), 0.0 µA (c)
and -0.8 µA (d).
the electron injection in the two arms of the QR.
Fig. 4.13 shines light on a difference behaviour between radial and
concentric fringes. In this figure, we study the effect of the probe current IDC , a DC current passing through the QR, applied in addition
to the AC current used to perform the conductance measurement. Increasing the absolute value of IDC is equivalent to raising the electron
excess energy relative to the Fermi energy. Fig. 4.13 (c) is the conductance map recorded at IDC = 0 µA, while Fig. 4.13 (b) and (d) were
recorded, respectively, for a large positive (+0.8 µA) and negative (0.8 µA) probe current, bringing the electron system out of thermal
equilibrium by more than 1 meV. The data can be better understood if
93
Chapter 4. Scanning Gate Microscopy of Quantum Rings
one divides the scan in three regions : the area enclosed in the QR where
radial features are predominant (I), the ring-shaped area in the vicinity
of the QR where concentric fringes dominate (II) and finally an area
situated far from the QR where the influence of the tip vanishes (III).
We first note that concentric fringes (region II) are strongly reduced
at large bias disrespect of its sign. Region I, on the other hand, exhibits a very different behaviour depending on current direction: while
radial fringes die out at large positive bias, they are strengthened and
rearranged when bias is reversed. A more quantitative picture of these
differences is presented in Fig. 4.13 (a), showing the evolution of δG,
the standard deviation of δG, calculated over regions I-III, as a function of IDC . As region III exhibits no dependence on IDC , it serves as a
reference for the noise level. Region II shows a decrease of δG at large
bias, which is symmetric in IDC . Since coherent effects are extremely
sensitive to electron excess energy, which enhances the electron-electron
scattering rates[82], this observation points towards a coherent origin
for the concentric fringes. By contrast, δG in region I shifts gradually
from one level to another as current is reversed.
For the time being, this asymmetric behaviour is not completely understood. We should, nevertheless, point out two aspects. Firstly, this
different behaviour indicates that there exist two perturbation regimes
in which the QR can be brought while being scanned by the tip.
Secondly, this set of images was obtained at 5.95 T. This constitutes the maximum magnetic field strength used in these experiments.
Furthermore, the presence of Shubnikov-de Haas oscillations (see section 3.4.1) shows that at this high magnetic field the cyclotron radius
is smaller than the dimensions of the ring and therefore the conduction
of the electrons through the ring is done, in part, by edge states.
In order to support better this argumentation, that radial and concentric fringes do not share the same origin, we do an analogous analysis using a further parameter: the tip-2DEG bias (Vtip ). In Fig. 4.14
we compare the evolution of the fringes amplitude in the different regions of the SGM images. Fig.4.14 (a) is a SGM image obtained for
Vtip = 2 V. As done previously, we distinguish three regions on the
SGM scans: region I is the area of the QR; region II corresponds to
the vicinity of the QR where concentric fringes are observed; region
III is situated in an area where the influence of the tip vanishes and is
used as reference. A quantitative comparison of the fringes behaviour
is shown in Fig. 4.14 (b) where we plot the standard deviation of the
conductance (δGSGM ) in each region as a function of Vtip . The fringes
amplitude in region I exhibits a fast quasi-linear increase with Vtip ,
while the fringes amplitude in region II remains essentially constant.
94
4.5. Experimental results
0.08
III
II
dGSGM[2e /h]
Region I
Region II
Region III
I
2
a
b
0
0.15
- 0.15
0.00
Vtip [V]
4
2
DGSGM[2e /h]
Figure 4.14: (a) Filtered SGM image of the conductance variations recorded
for Vtip = 2.5 V, Dtip = 50 nm, T = 4.2 K, B = 2 T. (b) Standard deviation
of the conductance in different regions versus tip bias. Region I corresponds
to the ring area. Region II corresponds to the fringes on both sides of the
ring. Region III is taken as reference.
4.5.3
Electrical AB effect in the SGM images
In this section we clarify the origin of the outside concentric fringes.
For this purpose we sweep the magnetic field in order to tune the phase
of interfering electrons.
Fig. 4.15 (a-c) show conductance maps measured at B = 1.5 T,
1.513 T and 1.526 T, covering a complete Aharonov-Bohm cycle, i.e.
the magnetic flux enclosed in the area of the QR changes by one flux
quantum. Qualitatively, the pattern of concentric and radial oscillations is surprisingly similar on this set of images. This apparent weak
sensitivity to magnetic field strongly contrasts with the results of experiments performed on large open quantum dots on the experiments
of Crook et. al.. This is a sign of the regular behaviour of the electron motion in QR, in opposition to the chaotic behaviour characterising large open quantum dots. In quantum rings, the magnetic field
acts in a more subtle way. This sequence of images reveal a continuous movement of fringes upon varying B. A quantitative analysis of
the concentric fringes evolution is obtained by computing the average
of the horizontal conductance profiles situated in regions α and β on
Fig. 4.15 (a) (delimited by the dashed lines). Fig. 4.15 (d-e) show the
evolution of these profiles over an Aharonov-Bohm cycle. On the left
of the QR (region α), the oscillation pattern smoothly moves leftwards
95
Chapter 4. Scanning Gate Microscopy of Quantum Rings
D
0.1
f0
f0 /2
Df = 0
b
a
b
a
c
d [mm]
0.00
d [mm]
0.25
d
0.50
0.00
a
e
0.25
0.50
b
0.0
Df [ f 0 ]
1.0
0.0
-0.1
D
-0.03
0.03
D
-0.03
0.03
Figure 4.15: Evolution of filtered conductance maps along an AB cycle. (ac) filtered conductance maps recorded at increasing magnetic fields between
B=1.5 T and 1.526 T (over one AB cycle), showing a continuous and mostly
periodic evolution of the pattern of fringes. In the vicinity of the ring, the
fringe pattern undergoes a cyclic evolution over the sequence: starting at
the position of a conductance maximum at the beginning of the cycle, one
observes a minimum at the middle of the cycle, and a conductance maximum
again at the end of the cycle. (d-e) average of horizontal conductance profiles
in region α and β (on both sides of the QR, defined by the dashed rectangles
on (a)), plotted as a function of the phase difference ∆φ, in units of the
Aharonov-Bohm period φ0 . The evolutions of the patterns are opposite in
the two regions (leftwards on (d), and rightwards on (e)), indicating that
the cyclic evolution of the fringes is not a measurement artefact due to a
movement of the QR during the measurement (the maximum value of this
movement corresponds to the error bar above (d)).
96
4.5. Experimental results
by one period as the Aharonov-Bohm phase - φ - changes by one flux
quantum : φ0 . Symmetrically, on the right of the QR (region β) the
oscillation pattern moves rightwards by one period as φ moves from 0
to φ0 (see dashed line on Fig. 4.15 (e)). We interpret this behaviour as
a scanning-gate induced electrostatic Aharonov-Bohm effect. Indeed,
as the tip approaches the QR, either from left or right, the electrical potential mainly raises on the corresponding side of the QR. This
induces a phase difference between electron wavefunctions travelling
through both arms of the ring, which causes the observed oscillations.
Since the magnetic field applies an additional phase shift to electron
wavefunctions, the V-shaped pattern on Fig. 4.15 (d-e) corresponds
to iso-phase lines for the electrons. This observation ensures that our
data are directly related to the Aharonov-Bohm effect (and not to a displacement of the QR during the AB cycle) and that concentric fringes
originate from an interference effect of coherent electrons. We finally
note that the observed oscillations are reminescent of those reported in
QRs with biased side gates [8][?]. However, our experiment brings direct spatial information on interference effects, as we take advantage of
the possibility to scan a much more localized tip-induced perturbation
across the sample.
Similar behaviour can be observed without applying a magnetic
field. In the previous section we compared the amplitude variation with
Vtip . Here we complement this observation. A closer inspection of these
images shows that the tip bias also affects the position of the outside
fringes. When Vtip is raised, the outside concentric fringes are shifted
away from the ring region. This effect is clearly evidenced in Fig.4.16(d)
and (e) where we plot the sequence of averaged profiles versus Vtip
for two regions (α and β) on both sides of the QR. These regions
were chosen for their fringes visibility. We interpret this behaviour as
the result of an electric field induced phase shift of the electron wave
functions. This phase shift changes the interference condition between
the electrons flowing in the two different arms of the ring, and thus the
transmission through the device.
The expansion of the outside fringes, combined with the insensitivity of their amplitude to the tip bias, are a clear indication that
they originate from a pure interference phenomenon. Therefore, in this
regime, we can directly compare the phase shifts induced by the tip
electric field and by the external magnetic field. In our experiment a
phase shift of π is obtained for a tip bias variation (∆Vtip ) of 1.75 V,
while using the magnetic field the same value is obtained with 13 mT.
Comparing both experiments we can estimate the lever arm of the
gating effect, i.e. the ratio between the electron gas potential and the
97
Chapter 4. Scanning Gate Microscopy of Quantum Rings
D
Vtip = 0.5 V
-0.15
0.0
0.15
Vtip = 3.5 V
Vtip = 2.0 V
a
b
a
d [mm]
0.00
d
0.45
0.00
a
e
d [mm]
0.45
b
0.5
DVtip=1.75 V
V tip [V]
3.5
c
b
D
-0.03
0.03
D
-0.03
0.03
Figure 4.16: Evolution of filtered conductance maps for different Vtip . (a-c)
SGM images for tip bias of 0.5 V, 2 V,and 3.5 V, respectively. (d-e) Sequence
of profile plots versus tip bias. Each horizontal line corresponds to a vertical
average of the conductance map in region α (d) and β (e), drawn on Fig.
3(c). The translation of the fringes to the left in (d) and to the right in (e)
is due to a pure electric Aharonov-Bohm effect.
applied tip voltage. In a simple model where a potential difference
∆Vgas is applied between the two arms of the QR of length L, using
Eq. 4.3 the phase shift ∆Φ of the electron wave functions writes:
∆Φ = π
e ∆Vgas L
EF λF
(4.4)
Using EF = 100 meV, λF = 20 nm, and L = 600 nm, which we believe to be close to experimental conditions, we obtain ∆Vgas = 3.3 mV
for a phase shift of π, and therefore a lever arm (∆Vgas /∆Vtip ) of about
0.002. This low value reflects a large electrostatic screening, due to the
large electron density in the 2DEG.
98
4.5. Experimental results
4.5.4
Imaging electronic wave function
To investigate it further on the SGM imaging mechanism, we compare experimental images with simulations. These simulations were
performed by Marco Pala in the scope of these experiments.
Indeed, the origin of the radial fringes observed within the QR area
can be better understood by means of quantum mechanical simulations
of the electron probability density at the Fermi energy in our QR [83].
Fig. 4.17 shows typical patterns of electron probability density |ψ 2 | obtained by summing the patterns obtained for each populated quantum
channel. While the precise position and number of fringes is sensitive
to the Fermi energy, simulations typically reveal radial fringes emerging
from the centre of the QR. Since the scattering probability of ballistic
electrons is proportional to the density probability, the conductance
of the QR should be mostly affected when the tip is located above a
region where |ψ 2 | is at maximum, and hence show radial fringes. The
typical conductance map shown in Fig. 4.18 is consistent with this picture: not only are there radial fringes, but their number and typical
lateral length scale are very similar to those in the electron probability
density computed for our QR. This similarity, together with the experimental data shown earlier, are clear indications that we are actually
locally probing electron flow as the tip of our microscope scans above
the quantum ring area.
Simulations
The device conductance and the wave functions at the Fermi energy are
calculated in the scattering matrix formalism and using the LandauerBüttiker formalism, with the same method as in [83]. The ring is
subdivided into slices perpendicular to axis x between both openings
in the QR. The Schrödinger equation is numerically solved in each slice
along y for each 1D channel and the plane wave solutions along x are
matched between each slice as prescribed in [84], giving the scattering
matrix between two slices. The overall scattering matrix and wave
functions are computed by composing the matrices of all slices. The
geometry (topology and dimensions) of the ring in the simulation is
comparable to that of our real sample, taking the depletion length at
device edges into account (∼35 nm), i.e. the inner and outer radii of
the ring are 140 nm and 265 nm, respectively, and the width of the
openings is 120 nm. The electron Fermi energy is adjusted close to
the value extracted from the 2DEG measurements, and the number of
quantum modes contributing to the QR conductance is chosen such
that the calculated conductance through the QR closely matches the
99
Chapter 4. Scanning Gate Microscopy of Quantum Rings
Figure 4.17: (a) Calculated conductance G vs Fermi energy EF in a QR with
inner and outer radii of 140 and 265 nm. (b-d) Electron probability density
Ψ2 (x, y, EF ) in a QR at EF = 104.6, 109.5 and 101 meV, respectively.
experimental value.
When EF is varied, the configuration of resonant states within the
QR and their coupling to the reservoirs change [74]. As a consequence,
the calculated conductance G exhibits fluctuations as a function of EF ,
one of the hallmarks of transport in open mesoscopic systems. This is
illustrated in Fig. 4.17(a), showing the calculated G as a function of EF
around the measured EF2DEG value in the unpatterned heterostructure.
Figs. 4.17(b-d) show typical patterns of |Ψ|2 (x, y, EF ) in our QR, for
EF = 101.0, 109.5 and 104.9 meV, respectively. Small-scale concentric
oscillations are visible within the whole QR area in Figs. 4.17(b-d),
whose characteristic spatial periodicity is related to the Fermi wavelength λF . On a scale larger than λF , |Ψ|2 (x, y, EF ) is rather homogeneous in Figs. 4.17(c-d), while it exhibits four strong radial fringes in
Figs. 4.17(b).
In analogy with the experiment, we have included in the simulation
a moving perturbation potential mimicking the tip effect, and then
calculated the conductance of the QR for each position of the perturbation. Fig. 4.18(c) shows such a simulated conductance map G(x, y),
obtained for EF = 101 meV, using the Gaussian potential V (x, y) as
100
4.5. Experimental results
Figure 4.18: (c-e) Simulated conductance maps G(x, y) for an electron probability density with a Fermi energy of EF = 101 meV (a). The gaussian
perturbing potential used to build each image is shown in (b). (c-e) are
obtain for Vm = EF /200 and σ = 5, 20 and 40 nm, respectively.
the moving perturbation,
(
"
(x − x0 )2 + (y − y0 )2
V (x, y) = Vm × exp −
2σ 2
#)
(4.5)
with Vm = EF /200, σ = 5 nm and (x0 , y0 ) the local position of the tip.
Most importantly, a careful examination of Fig. 4.18(d-e) reveals that
all the features visible in the simulated |Ψ|2 (x, y, EF ) are also visible in
the calculated G map. This striking correspondence reveals that SGM
can in principle be used to map the unperturbed electron probability
density.
Enlarging now the width of the perturbing potential causes the
smallest SGM features to disappear . As σ overcomes λF , concentric fringes completely vanish. Nevertheless, at the scale of the radial fringes, the correspondence between G(x, y) and |Ψ|2 (x, y, EF ) is
maintained (Fig. 4.18(d)) although the aspect of the four radial fringes
changes (Fig. 4.18(e)). Most importantly, the size of the smallest features in the simulated G maps is roughly correlated to σ. Based on
Figs. 4.18 (c-e), this allows us to infer a lower bound for σ ∼ 20-30 nm.
101
Distance [nm]
-80
1.5
80
6.2
b
d
5.9
1.0
c
5.6
0.5
0.0
b
c
0.15
2
d GSGM [2e /h ]
0
a
2
G [2e /h ]
Guassian Potential [meV]
Chapter 4. Scanning Gate Microscopy of Quantum Rings
d
Gaussian potential
Lorentzian potential
0.10
0.05
0.00
0.0
e
1.0
0.5
1.5
Vm [meV]
Figure 4.19: (a) Gaussian perturbing potential used to build simulated G
maps (c-e), i.e. calculated for EF = 101 meV, σ = 20 nm and Vm = 0.5,
1.0 and 1.5 meV, respectively. (e) Fringe amplitude δGSGM on simulated G
maps as a function of the maximum of gaussian potential (with σ = 20 nm)
and as a function of the maximum of Lorentzian potential.
To come closer to a complete description of our experiments, we now
examine the effect of the perturbation amplitude. As Vm is increased,
keeping σ = 20 nm, we observe that the SGM fingerprint remains qualitatively independent of Vm , and that its amplitude increases linearly
with Vm [Figs. 4.19(a-c)]; a behaviour consistent with the observations
related in Fig. 4.14. Above Vm ∼ 0.8 meV, qualitative changes in the
SGM pattern appear together with a deviation from the linear evolution of δGSGM vs Vm [Fig. 4.19(c)], a value for which differences between
G(x, y) and |Ψ|2 (x, y, EF ) start to emerge. Fig. 4.19(e) also shows that
the evolution of δGSGM vs Vm is weakly affected by the exact shape of
the perturbation, e.g. Lorentzian or Gaussian.
The consistency between the simulated behaviour of δGSGM vs Vm
and the experimental data in Fig. 4.14 leads us to conclude that, at low
Vtip (below ∼ 2 V), the central part of our SGM maps directly reveals
102
4.5. Experimental results
the main structure of |Ψ|2 (x, y, EF ) in our quantum rings.
This means that we can attribute the pattern of conductance fringes
in Fig. 4.14(a-c) and Fig. 4.19(c-d) to wavefunctions.
103
Chapter 4. Scanning Gate Microscopy of Quantum Rings
104
Chapter 5
Conclusion
In this thesis three aspects have taken our attention. The first concerns
the the AFM instrumentation. We have implemented a system adapted
to study electric properties of semiconductor structures. To do so, we
have chosen to work at 4 K and to have the possibility of working under
a magnetic field.
Working at low temperatures bring two advantages. The first one
concerns the resolution and stability
q of the instrument. The thermal
noise is decreased by a factor of 300/4 ≃ 8.7, which implies a gain
in force sensitivity and electrical charge detection sensitivity. Furthermore, working inside a cryostat, where the temperature gradients are
very small, limits the mechanical drift of piezoelectric elements. The
second reason is, indeed, mandatory. The coherent effects studied in
the Chapter 3 can only be observed at these low temperatures.
The second aspect studied in this thesis is related to charge detection. In Chapter 2, we presented a method to take advantage of a
parametric effect to measure electrical excess charges over a surface.
We have studied two particular situations. When the electric excitation is driven close to twice the natural frequency (ωel = 2ω0 + ∆ω)
and at ωel = ω0 + ∆ω. We have shown that in both cases a parametric
effect is present. It results from the fact that an alternative electrostatic force (ω) has, intrinsically, a component at 2ω. In this chapter we
presented an analytical model to show the presence of this effect. The
model was verified by numerical solutions and experiments. This analytical approach is particularly useful to derive the best measurement
conditions.
However, these techniques have not been implemented yet. Concerning the two approaches, the method ωel = 2ω0 + ∆ω seems to have
two significant advantages. Firstly, as shown in the same chapter, it can
beat the thermal noise limit. Secondly, in this situation the parametric
105
Chapter 5. Conclusion
effect intervenes in the form of a gain what may simplify the operation
of setting the system in the maximal sensitivity and the subsequent
interpretation of the experimental data.
This study made us conclude that, in contrast to a widespread belief, the ω component of the cantilever oscillation is not at all independent of the 2ω component of the electric force. This remark is particularly relevant in the case of ωel = ω0 + ∆ω which is largely known
in literature (KFM). We conclude that parametric effect prevails and
determines the oscillation amplitude whenever the tip gets close to the
sample.
In the last chapter we have addressed the electronic transport in
a model sample, the quantum ring. The most significant contribution
concerns the demonstration of the ability of mapping the electron wave
function inside a buried 2DEG. This constitute the most relevant result
of this thesis. We show that SGM may be seen as the analog of STM
for imaging the electronic the local density of states (LDOS) in open
mesoscopic systems buried under an insulating layer, or the counterpart
of the near-field scanning optical microscope that images the photonic
LDOS in confined nanostructures [85].
In our experimental conductance maps of the QR we observed reproducible structures. Using quantum mechanical simulations of the
electron probability density, including the perturbing potential of the
tip, we could reproduce the main experimental features and demonstrate the relationship between conductance maps and electron probability density maps.
Furthermore, this experiment clearly shows the capability of SGM
to image and to control electrons buried under a surface: in such an
experimental setup, the electron phase can be tuned using a flying gate.
This kind of instrument can therefore be used to modify in situ the
properties of coherent electronic devices.
We believe that, with this experiment, we showed that the combination of AFM with transport measurements is very powerful for
investigating electron interferences in real space at the local scale inside buried mesoscopic devices. One can envision to use the technique
to test electronic devices based on real-space manipulation of electron
interferences, such as the electronic analogs of optical or plasmonic
components (resonators, Y-splitters, ...). In situ control over the electron potential would provide a mean to design new ballistic devices with
desired characteristics (beam splitters, multiterminal devices...). Additionally, taking advantage of the controllable spin-orbit coupling effects
in InGaAs structures or other materials (InSb, magnetic semiconductors, ...), the technique may also serve to image the spin dynamics of
106
carriers within low-dimensional structures. Therefore, our study not
only provides a distinctive real-space visualization of the particle-wave
duality of electrons, but also paves the way for a wealth of experiments
probing the local behaviour of charge carriers inside a large variety of
open mesoscopic systems.
Nevertheless these themes are far from being exhausted. In this
thesis we only hope to have made a small contribution to this wide
research field.
107
Chapter 5. Conclusion
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